Rational series and their languages

Jean Berstel Christophe Reutenauer

Rational Series and Their Languages November 6, 2006

c 2006 Jean Berstel and Christophe Reutenauer

c English version Springer-Verlag 1988

c French edition Les s´eries rationnelles et leurs langages Masson 1984

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Preface to the electronic edition This electronic edition of the English edition is at the date of November 6, 2006, an almost literal transposition of the original text. The modifications concern notation, several proofs, one new statement, and some new exercises. It is planned that the electronic edition will progressively be modified, with inclusion of some new material, but will remain basically of the same size and of the same algebraic style. Notation Alphabets are named A, B, C, . . . instead of X, Y, Z, . . ., letters are a, b, c, . . . instead of x, y, z, . . .. Terminology prefix, suffix replaces left, right factor. New statements Corollary III.2.2 is a new statement. New exercises Exercise VIII.2.3 is new.

Marne-la-Vall´ee — Montr´eal Jean Berstel

November 6, 2006 Christophe Reutenauer

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Preface

Preface to the first English edition This book is an introduction to rational formal power series in several noncommutative variables and their relations to formal languages and to the theory of codes. Formal power series have long been used in all branches of mathematics. They are invaluable in enumeration and combinatorics. For this reason, they are useful in various branches of computer science. As an example, let us mention the study of ambiguity in formal grammars. It has appeared, for the past twenty years, that rational series in noncommutative variables have many remarkable properties which provide them with a rich structure. Knowledge of these properties makes them much easier to manipulate than, for instance, algebraic series. The depth and number of results for rational series are similar to those for rational languages. The aim of this text is to present the basic results concerning rational series. The point of view adopted here seems to us to be a natural one. Frequently one observes that a set of results becomes a theory when the initial combinatorial techniques are progressively replaced by more algebraic ones. We have tried wherever possible to substitute an algebraic approach for a combinatorial description. This has made it possible for us to give a unified and more complete presentation that is hopefully also easier to understand. We feel that, in this manner, the fundamental mechanisms and their interactions are easier to grasp. The first part of the book, comprising the first two chapters, illustrates very well how the introduction of an algebraic concept, namely syntactic algebra, can give a unified presentation. These two chapters contain the most important general results and discuss in particular the equality between rational and recognizable series and the construction of the reduced linear representation. The following two chapters are devoted to the two applications which seemed most important to us. First, we describe the relationship with the families of formal languages studied in theoretical computer science. Next, we establish the correspondence with the rational functions in one variable as studied in number theory. Chapter V presents arithmetic properties of rational series and their relations to the nature of their coefficients. These results are fairly profound, and there is a constant interaction with number theory. Let us mention the analytic characterization of N-rational series, which is the first result of this kind. The next chapter presents several results on decidability. We describe only some positive results which are of increasing importance. Those given here are v

vi

Preface

directly related to the Burnside problem. The last two chapters are devoted to the study of polynomials in noncommutative variables, and to their application to coding theory. Because of noncommutativity, the structure of polynomials is much more complex that it would be in the case of commutativity, and the results are rather delicate to prove. We present here basic properties concerning factorizations, without trying to be complete. The main purpose of Chapter VII is to prepare the ground for the final chapter which contains the generalization of a result of M.-P. Sch¨ utzenberger concerning the factorization of a polynomial associated with a finite code. Exercises are provided for most chapters and also short bibliographical notes. The algebraic and arithmetic approach adopted in this book implies a choice in the set of possible applications. We do not describe several important applications, such as the use of polynomials in control theory, where formal series in noncommutative variables are employed to represent the behavior of systems and replace the Volterra series (Fliess 1981, Isidori 1985). Another area of application is combinatorial graph theory. Enumeration of graphs by wellchosen encodings leads to systems of equations in noncommutative formal series whose solutions give the desired enumeration. Cori (1975) gives an introduction to the topic. The analysis of algorithms also leads to the study of formal series in a somewhat larger context (see Steyaert and Flajolet 1983, Berstel and Reutenauer 1982). This book issued from an advanced course held several times by the authors, at the University Pierre et Marie Curie, Paris and at the University of Saarbr¨ ucken. Parts of the book were also taught at several different levels at other places. Any concept from algebra that might not be familiar to the reader can be found in S. Lang’s Algebra (Lang 1984). Finally, thanks are due to Rosa de Marchi who carefully typed the manuscript. Paris — Montr´eal August 1988

Jean Berstel Christophe Reutenauer

Note to the reader Following usual notation, items such as sections, theorems, corollaries, etc. are numbered within a chapter. When cross-referenced the chapter number is omitted if the item is within the current chapter. Thus “Theorem 1.1” means the first theorem in the first section of the current chapter, and “Theorem II.1.3” refers to the equivalent theorem in Chapter II. Exercises are numbered accordingly and the section number should help the reader to find the section relevant to that exercise.

Contents Chapter I Rational Series 1 Semirings . . . . . . . . . . 2 Formal Series . . . . . . . . 3 Topology . . . . . . . . . . 4 Rational Series . . . . . . . 5 Recognizable Series . . . . . 6 The Fundamental Theorem Exercises . . . . . . . . . . Notes . . . . . . . . . . . .

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1 1 2 3 4 7 12 15 19

Chapter II Minimization 1 Syntactic Ideals . . . . . . . . . . 2 Reduced Linear Representations 3 The Reduction Algorithm . . . . Exercises . . . . . . . . . . . . . Notes . . . . . . . . . . . . . . .

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21 21 26 29 34 34

Chapter III Series and Languages 1 The Theorem of Kleene . . . . 2 Series and Rational Languages 3 Support . . . . . . . . . . . . . 4 Iteration . . . . . . . . . . . . . 5 Complementation . . . . . . . . Exercises . . . . . . . . . . . . Notes . . . . . . . . . . . . . .

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35 35 37 40 43 44 47 48

Chapter IV Rational Series in One Variable 1 Rational Functions . . . . . . . . . . 2 The Exponential Polynomial . . . . 3 A Theorem of P´ olya . . . . . . . . . 4 A Theorem of Skolem, Mahler, Lech Exercises . . . . . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . .

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49 49 54 58 62 70 70

Chapter V Changing the Semiring 1 Rational Series over a Principal Ring . . . . . . . . . . . . . . . . 2 Positive Rational Series . . . . . . . . . . . . . . . . . . . . . . . 3 Fatou Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . .

71 71 74 83

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viii

Contents Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Chapter VI Decidability 1 Problems of Supports 2 Growth . . . . . . . . Exercises . . . . . . . Notes . . . . . . . . .

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89 89 91 96 96

Chapter VII Noncommutative polynomials 1 The Weak Algorithm . . . . . . . . 2 Continuant Polynomials . . . . . . 3 Inertia . . . . . . . . . . . . . . . . 4 Gauss’s Lemma . . . . . . . . . . . Exercises . . . . . . . . . . . . . . Notes . . . . . . . . . . . . . . . .

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99 99 102 106 111 113 114

Chapter VIII Codes and Formal Series 1 Codes . . . . . . . . . . . . . 2 Completeness . . . . . . . . . 3 The Degree of a Code . . . . 4 Factorization . . . . . . . . . Exercises . . . . . . . . . . . Notes . . . . . . . . . . . . .

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115 115 119 123 124 131 132

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References

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Index

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Chapter I

Rational Series This chapter contains the definitions of the basic concepts, namely rational and recognizable series in several noncommutative variables. It also gives a short account of some preliminary notions that will appear frequently throughout the book. We start with the definition of a semiring, followed by the notation for the usual objects in free monoids and formal series. The topology on formal series is only treated to the extent required for later reference. Section 4 contains the definition of rational series, together with some elementary properties and the fact that certain morphisms preserve the rationality of series. Recognizable series are introduced in Section 5. An algebraic characterization is given. We also prove (Theorem 5.1) that the Hadamard product preserves recognizability. The fundamental theorem of Sch¨ utzenberger (equivalence between rational and recognizable series, Theorem 6.1) is the concern of the last section. This theorem is the starting point for the developments given in the subsequent chapters.

1

Semirings

Recall that a monoid is a set equipped with an associative binary operation and having a neutral element for this law. A semiring is, roughly speaking, a ring without subtraction. More precisely, it is a set K equipped with two operations + and · (sum and product) such that the following properties hold: (i) (ii) (iii) (iv)

(K, +) is a commutative monoid with neutral element denoted by 0. (K, ·) is a monoid with neutral element denoted by 1. The product is distributive with respect to the sum. For all a in K, 0a = a0 = 0.

The last property is not a consequence of the others, as is the case for rings. A semiring is commutative if its product is commutative. A subsemiring of K is a subset of K containing 0 and 1, which is stable for the operations of K. 1

2

Chapter I. Rational Series A semiring morphism is a function f : K → K′

of a semiring K into a semiring K ′ that maps the 0 and 1 of K into the corresponding elements of K ′ and that respects sum and product. Let us give some examples of semirings. Among them are, of course, fields and rings. Next, the set N of natural numbers, the sets Q+ of nonnegative rational numbers and R+ of nonnegative real numbers are semirings. The Boolean semiring B = {0, 1} is completely described by the relation 1 + 1 = 1 (see Exercise 1.1). If M is a monoid, the set of its subsets is naturally equipped with the structure of a semiring: the sum of two subsets X and Y of M is simply X ∪ Y and their product is {xy | x ∈ X, y ∈ Y } . Let K be a semiring and let P, Q be two finite sets. We denote by K P ×Q the set of P × Q-matrices with coefficients in K. The sum of such matrices is defined in the usual way, and if R is a third finite set, a product K P ×Q × K Q×R → K P ×R is defined in the usual manner. In particular, K Q×Q thus becomes a semiring. If P = {1, . . . , m} and Q = {1, . . . , n}, we will write K m×n for K P ×Q ; moreover, K 1×1 will be identified with K. For the rest of this chapter, we fix a semiring K.

2

Formal Series

Let A be a finite, nonempty set called alphabet . The free monoid A∗ generated by A is the set of finite sequences a1 · · · an of elements of A, including the empty sequence denoted by 1. This set is a monoid, the product being the concatenation defined by (a1 · · · an ) · (b1 · · · bp ) = a1 · · · an b1 · · · bp and with neutral element 1. An element of the alphabet is called a letter , an element of A∗ is a word , and 1 is the empty word . The length of a word w = a1 · · · an is n; it is denoted by |w|. The length |w|a relative to a letter a is defined to be the number of occurrences of the letter a in w. We denote by A+ the set A∗ \ 1. A language is a subset of A∗ . A formal series (or formal power series) S is a function A∗ → K . The image by S of a word w is denoted by (S, w) and is called the coefficient of w in S. The support of S is the language supp(S) = {w ∈ A∗ | (S, w) 6= 0} .

3

3. Topology

The set of formal series over A with coefficients on K is denoted by KhhAii. A structure of a semiring is defined on KhhAii as follows. If S and T are two formal series, their sum is given by (S + T, w) = (S, w) + (T, w) , and their product by X (ST, w) = (S, x)(T, y) . xy=w

Observe that this sum is finite. Furthermore, two external operations of K on KhhAii, one acting on the left, the other on the right, are defined, for k ∈ K, by (kS, w) = k(S, w),

(Sk, w) = (S, w)k .

There is a natural injection of the free monoid into KhhAii as a multiplicative submonoid; the image of a word w is still denoted by w. Thus the neutral element of KhhAii for the product is 1. Similarly, there is an injection of K into KhhAii as a subsemiring: to each k ∈ K is associated k · 1 = 1 · k, simply denoted by k. Thus we identify A∗ and K with their images in KhhAii. A polynomial is a formal series with finite support. The set of polynomials is denoted by KhAi. It is a subsemiring of KhhAii. The degree of a polynomial is the maximal length of the words in its support (and is −∞ if the polynomial is zero). When A = {a} has just one element, one get the usual sets of formal power series Khhaii = K[[a]] and of polynomials Khai = K[a]. For the rest of this chapter, we fix an alphabet A.

3

Topology

We have seen that KhhAii is the set of functions A∗ → K. In other words, ∗

KhhAii = K A . Thus, if K is equipped with the discrete topology, the set KhhAii can be equipped with the product topology. This topology can be defined by an ultrametric distance. Indeed, let ω : KhhAii × KhhAii → N ∪ ∞ be the function defined by ω(S, T ) = inf{n ∈ N | ∃w ∈ A∗ , |w| = n and (S, w) 6= (T, w)} . For any real number σ with 0 < σ < 1, the function d : KhhAii × KhhAii → R d(S, T ) = σ ω(S,T )

4

Chapter I. Rational Series

is an ultrametric distance, that is d is a distance which satisfies the enforced triangular inequality d(S, T ) ≤ max(d(S, U ), d(U, T )) The function d defines the topology given above (Exercise 3.1). Furthermore, KhhAii is complete for this topology, and it is a topological semiring (that is sum and product are continuous functions). Let (Si )i∈I be a family of series. It is called summable if there exists a formal series S such that for all ε > 0, there exists a finite subset I ′ if I such that every finite subset J of I containing I ′ satisfies the inequality X  d Sj , S ≤ ε . j∈J

The series S is then called the sum of the family (Si ) and it is unique. A family (Si )i∈I is called locally finite if for every word w there exists only a finite number of indices i ∈ I such that (Si , w) 6= 0. It is easily seen that every locally finite family is summable. The sum of such a family can also be defined simply for w ∈ A∗ by X (S, w) = (Si , w) , i∈I

observing that the support of this sum is finite because the family (Si ) is locally finite (all terms but a finite number in this sum are 0). However, it is not true that a summable family is always locally finite (see Exercise 3.2), but we shall need mainly the second concept. Let S be a formal series. Then the family of series ((S, w)w)w∈A∗ clearly is locally finite, since each of these series has a support formed of at most one single word, and supports are pairwise disjoint. Thus the family is summable, and its sum is just S. This gives the usual notation X S= (S, w)w . w∈A∗

It follows in particular that KhAi is dense in KhhAii which thus is the completion of KhAi for the distance d.

4

Rational Series

A formal series S ∈ KhhAii is proper if the coefficient of the empty word (that is the constant term of S) vanishes, thus if (S, 1) = 0. In this case, the family (S n )n≥0 is locally finite. Indeed, for any word w, the condition n > |w| implies (S n , w) = 0. Thus the family is summable. The sum of this family is denoted by S ∗ X S∗ = Sn , n≥0

and is called the star of S. Similarly, S + denotes the series X S+ = Sn . n≥1

5

4. Rational Series

The fact that KhhAii is a topological semiring and the usual properties of summable families imply that S∗ = 1 + S+

and S + = SS ∗ = S ∗ S .

From these, it follows that if K is a ring, then S ∗ is just the inverse of 1 − S since S ∗ (1 − S) = S ∗ − S ∗ S = S ∗ − S + = 1. This also implies the following classical result: a series is invertible if an d only if its constant term is invertible in K (still assuming K to be a ring); see Exercise 4.5. Let us return to the general case of a semiring. Lemma 4.1 Let T and U be formal series, with T proper. Then the unique solution S of the equation S = U + T S (of S = U + ST ) is the series S = T ∗ U (the series S = U T ∗ , respectively). Proof. One has T ∗ = 1 + T T ∗ , whence T ∗ U = U + T T ∗U . Conversely, since T is proper X lim T n = 0 and lim Ti = T∗ . n

n

0≤i≤n

From S = U + T S, it follows that S = U + T (U + T S) = U + T U + T 2 S and inductively S = (1 + T + · · · + T n )U + T n+1 S . Thus, going to the limit, and using the fact that KhhAii is a topological semiring, one gets S = T ∗ U .  Definition The rational operations in KhhAii are the sum, the product, the two external products of K on KhhAii and the star operation. A subset of KhhAii is rationally closed if it is closed for the rational operations. The smallest subset containing a subset E of KhhAii and which is rationally closed is called the rational closure of E. Definition A formal series is rational if it is in the rational closure of KhAi. Observe that if K is a ring, then the rational closure of KhAi is the smallest subring of KhhAii containing KhAi and closed under inversion (in other words, the star operation and inversion play equivalent roles). Definition If L is a language, its characteristic series is the formal series X w. L= w∈L

In other words, (L, w) = 1 for w ∈ L, and (L, w) = 0 if w ∈ / L.

6

Chapter I. Rational Series

Example 4.1 The series A is proper and X An . A∗ = n≥0

n

Since A is the sum of all words of length n, it follows that X A∗ = w w∈A∗

is the characteristic series of A∗ . Thus, this series is rational. Consider now a letter a. The series A∗ aA∗ , as a product of A∗ , a, and A∗ , is also rational. By the definition of product, X (A∗ aA∗ , w) = (A∗ , x)(a, y)(A∗ , z) . xyz=w

Since (a, y) = 0 unless y =P a (and then (a, y) = 1), and since (A∗ , x) = (A∗ , z) = ∗ ∗ 1, one has (A aA , w) = xaz=w 1, which is the number of factorizations w = xaz, that is the number |w|a of occurrences of the letter a in w. Thus X A∗ aA∗ = |w|a w w

is a rational series. Let B be an alphabet, and let ρ be a function ρ : A → KhhBii . Then ρ extends to a morphism of monoids ρ : A∗ → KhhBii . If K is commutative, then ρ can be extended in a unique manner into a morphism of semirings ρ : KhAi → KhhBii with ρ|K = id. Indeed, it suffices, for any polynomial P = KhAi, to set X ρ(P ) = (P, w)ρ(w)

P

w∈A∗ (P, w)w

∈

w∈A∗

which is a finite sum since P is a polynomial. Then ρ isK-linear. Moreover, in view of the commutativity of K X X ρ(P )ρ(Q) = (P, x)ρ(x) (Q, y)ρ(y) x∈A∗

X

=

y∈A∗

(P, x)ρ(x)(Q, y)ρ(y) =

x,y∈A∗

=

X

X

x,y∈A∗

(P, x)(Q, y)ρ(xy)

x,y∈A∗

=ρ

 X

x,y∈A∗

 (P, x)(Q, y)xy = ρ(P Q) .

(P, x)(Q, y)ρ(x)ρ(y)

7

5. Recognizable Series

Assume now that for each letter a ∈ A, the series ρ(a) is proper. Then ρ : KhAi → KhhBii is uniformly continuous. Indeed, let P and Q be two polynomials with ω(P, Q) = n . Then, for any word x in B ∗ of length < n, X X (P, w)(ρ(w), x) (ρ(P ), x) = (P, w)(ρ(w), x) = w∈A∗

|w| 0, then k = p − q + 1 and q + k = p + 1. In all cases, q + k > p. Since QS = P is a polynomial of degree deg(P ), one has, for all n ∈ N, 0 = (P, xn+q+k ) = an+q+k + β1 an+q+k−1 + · · · + βq an+k . Thus, since Q(x) = xq + β1 xq−1 + · · · + βq , S ◦ (xk Q) = 0 .

52

Chapter IV. Rational Series in One Variable

This shows that xk Q is in the syntactic ideal of S, and consequently R divides xk Q. Thus r ≤ q + k. If k = 0, then r ≤ q, q ≤ s and s ≤ r imply that all these numbers are equal, whence R = Q and Q = R. If k 6= 0, then k = deg P − deg Q + 1, and since P1 Q = P R, k = deg P1 − deg R + 1 ≤ r − deg R , whence k + q ≤ k + s ≤ r. Thus r = k + q and s = q, showing that R = xk Q and Q = R.  P The Hankel matrix of S = an xn has a very special form, which is classical. It is the matrix (ai+j )i,j∈N . P Corollary 1.3 Let S = an xn be a rational series with associated irreducible fraction P (x)/Q(x). Its rank is equal to sup(deg Q, 1 + deg P ), to the degree of its minimal polynomial, to the length of the shortest linear recurrence relation satisfied by S, and to the rank of its Hankel matrix. Proof. We have only to verify the rank property. We take the notations of the previous proof. If k = 0, then p < q and the rank is deg(R) = q = sup(q, p + 1). If k > 0, then k = p − q + 1 and deg(R) = k + deg(Q) = k + deg(Q) = p + 1 = sup(q, p + 1), since p − q + 1 > 0.  Observe that the set of eigenvalues 6= 0 of S is precisely the set of inverses of its poles, with the same multiplicities. Definition A rational series is regular if it admits a linear representation (λ, µ, γ) such that µx is an invertible matrix. Regular rational series can be defined inP several ways. Indeed, the following assertions concerning a rational series S = an xn are equivalent. (i) S is regular. (ii) Any reduced linear representation (λ, µ, γ) of S is regular , that is the matrix µx is invertible. (iii) The sequence (an ) satisfies a proper linear recurrence relation, that is an+k = α1 an+k−1 + · · · + αk an , (iv) (v) (vi) (vii)

n ≥ 0, αk 6= 0 .

The shortest linear recurrence relation satisfied by S is proper. There exists a polynomial P such that S ◦ P = 0 and P (0) 6= 0. The minimal polynomial of S has a non vanishing constant term. S = P (x)/Q(x) with deg P < deg Q.

The equivalence of these assertions is a consequence of the preceding propositions and of the following observation: if (an ) satisfies some proper linear recurrence relation and if m is the the companion matrix of this relation, then det(m) 6= 0 and there exist λ, γ such that an = λmn γ. Proposition 1.4 For every rational series S, there exist a unique couple (T, P ), where T is a regular series and P is a polynomial, such that S = P + T .

1. Rational Functions

53

This proposition is a direct consequence of the decomposition of the rational fraction associated with S into simple elements. Then P is just the integral part of the fraction. We give here a different proof. Observe that, as a consequence of this result, a regular rational series which is a polynomial is null. Proof. Let xq R(x), with R(0) 6= 0, be the minimal polynomial of S. Then (S ◦ R) ◦ xq = S ◦ (xq R) = 0

which shows that S ◦ R is a polynomial. Consider the function Q 7→ Q ◦ R K[x] → K[x] Since R(0) 6= 0, one has deg(Q ◦ R) = deg(Q), and this function is consequently a linear automorphism of K[x]. Thus there is some P in K[x] such that P ◦R = S ◦R. Let T = S − P . Then T ◦R = S ◦R −P ◦R = 0, showing that T is regular rational. If T + P = T ′ + P ′ , where T and T ′ are regular rational series and P, P ′ are polynomials, then T − T′ = P′ − P

In view of condition (vii) above, the series T − T ′ is regular. Thus it suffices to show P that if S is regular and is a polynomial, then S = 0. For this, set S = an xn . There exist coefficients αi in K such that for all n ≥ 0 an+k = α1 an+k−1 + · · · + αk an

(1.2)

with αk = 6 0. Assume S 6= 0, and let n be the greatest index such that an 6= 0. For this n, Eq. (1.2) gives αk an = 0, whence an = 0, a contradiction.  In view of Proposition 1.4, it suffices for many purposes to study regular rational series. We will restrict ourselves to these series in the following. Proposition 1.5 The subset of regular rational series of K[[x]] is closed under linear combination, product, and Hadamard product. Observe that this set does not contain any non vanishing polynomials. Proof. Let S1 = P1 /Q1 and S2 = P2 /Q2 be regular series with deg(P1 ) < deg(Q1 ) and deg(P2 ) < deg(Q2 ). Then S1 + S2 = (P1 Q2 + P2 Q1 )/Q1 Q2 and S1 S2 = P1 P2 /Q1 Q2 . Since deg(P1 Q2 + P2 Q1 ) < deg(Q1 Q2 ) and deg(P1 P2 ) < deg(Q1 Q2 ), the series S1 + S2 and S1 S2 are regular. Moreover, if (S1 , xn ) = λ1 µ1 xn γ1 and (S2 , xn ) = λ2 µ2 xn γ2 , where µ1 x and µ2 x are invertible matrices, then (S1 ⊙ S2 , xn ) = (S1 , xn )(S2 , xn ) = (λ1 ⊗ λ2 )(µ1 ⊗ µ2 )(xn )(γ1 ⊗ γ2 ) , and since (µ1 ⊗ µ2 )(x) is invertible, this shows that S1 ⊙ S2 is regular.  The set of regular rational series equipped with the structure of vector space and with the Hadamard product is thePHadamard algebra of regular rational series. Its neutral element is the series xn = 1/(1 − x).

54

2

Chapter IV. Rational Series in One Variable

The Exponential Polynomial

We assume from now on that K has characteristic zero. Let Λ be the multiplicative group K \ 0, and let t be an indeterminate. We consider the algebra K[t][Λ] of the group Λ over the ring K[t]. It is in particular an algebra over K. An element of K[t][Λ] is called an exponential polynomial . Theorem 2.1 Let K be algebraically closed. The function which associates to an exponential polynomial X Pλ (t)λ λ∈Λ

of K[t][Λ] the regular rational series X an xn n≥0

defined by an =

X

Pλ (n)λn

λ∈Λ

(with the sum computed in K) is an isomorphism of K-algebra from K[t][Λ] onto the Hadamard algebra of regular rational series. P Proof. Let φ be the function of the statement. Let E = PλP (t)λ and F = P Qλ (t)λ be two exponential polynomials, and let G = E +F = Rλ (t)λ, H = P EF = Sλ (t)λ ∈ K[t][Λ]. Then X Rλ = Pλ + Qλ , Sλ = Pµ Qν . µν=λ

Consequently (φ(G), xn ) =

X

X

Pλ (n)λn +

λ

µν=λ

X

Qν (n)ν n

Rλ (n)λn =

X

Qλ (n)λn

= (φ(E), xn ) + (φ(F ), xn ) , X X X (φ(H), xn ) = Sλ (n)λn = λn Pµ (n)Qν (n) =

X

Pµ (n)µn

µ

ν

n

= (φ(E), x )(φ(F ), xn ) .



Thus φ(E + F ) = φ(E) + φ(F ), φ(EF ) = φ(E)φ(F ) . Let us now verify that φ is a bijection. Let α1 , . . . , αk be elements of K with P αk 6= 0, and let V be the set of all (regular rational) series S = an xn satisfying the relation an+k = α1 an+k−1 + · · · + αk an ,

(n ≥ 0) .

2. The Exponential Polynomial

55

Clearly, V is a vector space of dimension k. Let λ1 , . . . , λp be the roots of the polynomial R(x) = xk − α1 xk−1 − · · · − αk with multiplicities n1 , . . . , np respectively. Consider the subspace V ′ of K[t][Λ] of dimension k o n X Pi (t)λi | deg(Pi ) ≤ ni − 1 V′ = 1≤i≤p

We show that φ induces a surjection V ′ → V (and consequently an injection) and this will P prove the theorem. Any S = an xn in V can be written as P (x)/Q(x), with deg(P ) < deg(Q) and Q being the reciprocal polynomial of R. Decomposing P/Q into simple elements shows that S is a linear combination of series 1 , (1 − λi x)j

1 ≤ i ≤ p, 1 ≤ j ≤ nj .

Next, it is well-known that X  n + j − 1 1 = λn xn . (1 − λx)j j−1 n≥0


Since n+j−1 is a polynomial of degree j − 1 in the variable n, the surjectivity j−1 of φ : V ′ → V is proved.  Observe that in the bijection described P in the theorem and its proof, the support of an exponential polynomial E = Pλ (t)λ (that is the set of λ ∈ Λ such that Pλ 6= 0) is exactly the set of eigenvalues (that is inverses of poles) of S, and that the multiplicity of a eigenvalue λ is equal to 1 + deg(Pλ ). Furthermore, if the coefficients and the eigenvalues of S are in some subfield K1 of K, then the corresponding exponential polynomial is in K1 [t][Λ1 ], with Λ1 = K1 \ 0. P Corollary 2.2 Let S = an xn be a rational series over an algebraically closed field K of characteristic 0. (i) The coefficients an are given, for large enough n, by X an = λni Pi (n) ,

(2.1)

1≤i≤p

where λ1 , . . . , λp ∈ K \ 0 and Pi (t) ∈ K[t]. (ii) The expression (2.1) is unique if the λi ’s are distinct; in particular, the nonzero eigenvalues of S are the λi ’s with Pi 6= 0. Proof. (i) By Proposition 1.4, S = P + T for some polynomial P and some rational regular series T . Thus, it suffices to use Theorem 2.1. (ii) Let  X X T = λni Pi (n) xn n≥0 1≤i≤p

56

Chapter IV. Rational Series in One Variable

Then, in view of Theorem 2.1, T is rational regular. Moreover S = P + T for some polynomial P (because S and T have by assumption the same coefficients for large enough n). By Proposition 1.4, T depends only on S, and by Theorem 2.1, the exponential polynomial of T is unique. This proves the first assertion. By the remark following the proof of Theorem 2.1, the λi ’s with Pi 6= 0 are exactly the eigenvalues of T . Now, it is clear that T and S have the same poles, so they have the same nonzero eigenvalues.  Definition Let S0 , . . . , Sp−1 be formal series in K[[x]]. The merge of these series is the formal series defined for m ∈ N and i ∈ {0, . . . , p − 1} by (S, xmp+i ) = (Si , xm ) . In other words, if n = mp + i (Euclidean division of n by p), then (S, xn ) = (Si , xm ). This can also be written as X

S(x) =

xi Si (xp )

0≤i s), then v(aj+qn ) = v(bj+qn ) = s . Thus it suffices to set n0 = j + qn′ , where n′ is chosen so that n0 > r. This proves the preliminary claim. (ii) The series S is rational over Q. We may assume that it is regular by Proposition 1.4. By Exercise I.5.1.b, we may assume that it is rational over Z and has a linear representation (λ, µ, γ) with µx over Z and of nonzero determinant. Let P (x) = xr − α1 xr−1 − · · · − αr be its characteristic polynomial. Then (an ) satisfies the linear recurrence relation associated to P . The roots λ1 , . . . , λk of P are algebraic integers. Let K be the number field K = Q[λ1 , . . . , λk ]. By Theorem 2.1, the an admit the expression given by Eq. (3.1). Moreover, for any prime ideal p of K, the αi and an are in the valuation ring of K for the valuation vp and by our preliminary result (i), there exist integers j and ℓ such that n 7→ vp (aj+ℓn ) is an affine function. (iii) Let B be the ring of algebraic integers of K, and let p be a prime number. The ideal pB of B decomposes as ms 1 pB = pm 1 · · · ps ,

where p1 . . . , ps are distinct prime ideals of K. By applying the preceding argument for p = p1 one obtains integers j, ℓ such that the function n 7→ vp1 (aj+ℓn ) is affine. By iteration of this computation for p2 , . . . , ps , one gets successive subsequences and finally one obtains an arithmetic progression n′0 + q ′ N such that for each i = 1, . . . , s, the function n 7→ vpi (an′0 +q′ n )

61

3. A Theorem of P´ olya is affine. Thus there exist integers xi and yi such that vpi (an′0 +q′ n ) = xi + yi n .

Note that xi , yi are integers, since xi + yi n is an integer for n in N. Now observe that for all a ∈ Z,    vpi (a) ; i = 1, . . . , s vp (a) = inf mi where ⌊z⌋ denotes the integral part of z. Since the functions n 7→

vpi (an′0 +q′ n ) xi + yi n = mi mi

also are affine, there exists an integer i0 such that for all i = 1, . . . , s and all sufficiently large n, 1 1 (xi + yi n) ≥ (xi + yi0 n) , mi mi0 0 showing that 

xi0 + yi0 n vp (an′0 +q′ n ) = mi0



for sufficiently large n. Since the function     xi0 + yi0 mi0 n xi0 n 7→ = + yi0 n mi0 mi0 also is affine, the lemma follows.  Proof of Theorem 3.1. Let S be a rational series having a finite set of prime factors. Clearly we may assume that S is regular (Proposition 1.4). In view of Proposition 2.4, P we nmay even assume that S is simple. Let S = an x and let p1 , . . . , pℓ be the prime factors of S. Applying Lemma 3.2 successively to p1 , . . . , pℓ , one obtains integers n0 and q such that, for every i = 1, . . . , ℓ, the function n 7→ vpi (an0 +qn ) is affine. Set ǫk = −1, 0, 1 according to an < 0, an = 0, an > 0. Then for n ≥ 0, one has an0 +qn = θn bcn with θn = ǫn0 +qn . Now let λ1 , . . . , λk , with k ≥ 1 be the distinct eigenvalues of S. In view of Theorem 2.1, there are non vanishing polynomials P1 . . . , Pk such that an =

k X i=1

Pi (n)λni .

(3.2)

62

Chapter IV. Rational Series in One Variable

Thus, setting bn = an0 +qn , Qi (t) = Pi (n0 + qt)λni 0 , µi = λqi , one has bn = θn bcn =

k X

Qi (n)µni .

i=1

Since the group generated by the λi ’s is free, P all the µi are distinct. Moreover, the polynomials Qi (t) do not vanish, and thus bn xn is not a polynomial. Thus θn 6= 0 for infinitely many n, and we may suppose that θn = 1 for infinitely many n. The series X bn

cn

xn

has finite image. By Theorem III.2.8 (and Exercise III.1.1), there exists an arithmetic progression n1 + rN such that θn = 1 for n ∈ n1 + rN. Thus bn1 +rn = bcn1 (cr )n =

k X

Qi (n1 + rn)µn1 (µri )n .

i=1

As before, the µri are pairwise distinct. In view of the unicity of the exponential polynomial, one has k = 1 and Q1 (n1 + rt) = C, for some constant. Thus Q1 is a constant and also P1 . By Eq. (3.2), an = P1 λn1 . This completes the proof. 

4

A Theorem of Skolem, Mahler, Lech

The following result describes completely the supports of rational series in one variable with coefficients in a field of characteristic zero. They are exactly the rational one-letter languages. This does not hold for more than one variable (see Example III.3.1). Theorem 4.1 (Skolem 1934, 1935, Lech 1953) Let K be a field of charP Mahler acteristic 0, and let S = an xn be a rational series with coefficients on K. The set {n ∈ N | an = 0} is the union of a finite set and of a finite number of arithmetic progressions. In fact, this result has been proved for K = Z by Skolem, it has been extended to algebraic number fields by Mahler and to fields of characteristic 0 by Lech. This author also gives the following example showing that the theorem does not hold in characteristic p 6= 0. P Indeed, let θ be transcendent over the field Fp with p elements. Then the series an xn with an = (θ + 1)n − θn − 1

is rational over Fp (θ) and, however, {n | an = 0} = {pr | r ∈ N} is not a rational subset of tne monoid N.

4. A Theorem of Skolem, Mahler, Lech

63

The proof given here is elementary and does not use p-adic analysis. It requires several definitions and lemmas, and goes through three steps. First, the result is proved for series with integral coefficients. Then it is extended to transcendental extensions and finally to the general case. Definitions A set A of nonnegative integers is called purely periodic if there exist an integer N ≥ 0 and integers k1 , k2 , . . . , kr ∈ {0, 1, . . . , N − 1} such that A = {ki + nN | n ∈ N, 1 ≤ i ≤ r} . The integer N is a period of A. A quasi-periodic set (of period N ) is a subset of N which is the union of a finite set and of a purely periodic set (of period N ). Lemma 4.2 The intersection of a family of quasi-periodic sets of period N is quasi-periodic of period N . Proof. Let (Ai )i∈I be a family of quasi-periodic sets, all having period N . Given a j ∈ {0, 1, . . . , N − 1}, for any i ∈ I, the set (j + N N) ∩ Ai is either finite or equal to j + N N. Thus the same holds for (j + N N) ∩ (∩Ai ).  Definition Given a series S = annihilator of S is the set

P

an xn with coefficients in a semiring K, the

ann(S) = {n ∈ N | an = 0} . Thus the annihilator is the complement of the support. With these definitions, the first (and most difficult) step in the proof of Theorem 4.1 can be formulated as follows. P Proposition 4.3 Let S = an xn ∈ Q[[x]] be a regular rational series with rational coefficients. Then the annihilator of S is quasi-periodic. Let p be a fixed prime number. The p-adic valuation vp is defined at the beginning of Section 3. Observe that X vp (q1 · · · qn ) = vp (qi ) 1≤i≤n

vp (q1 + · · · + qn ) ≥ inf{vp (q1 ), . . . , vp (qn )} .

Observe also that for n ∈ N vp (n!) ≤ n/(p − 1) since indeed (Exercise!) vp (n!) = ⌊n/p⌋ + ⌊n/p2 ⌋ + · · · + ⌊n/pk ⌋ + · · · ≤ n/p + n/p2 + · · · + n/pk + · · · X 1 1/p =n = n/(p − 1) . ≤n pk 1 − 1/p k≥1

(4.1)

64

Chapter IV. Rational Series in One Variable

From Eq. (4.1), we deduce  n p n vp = vp (pn ) − vp (n!) ≥ n − , n! p−1 thus vp



pn n!



≥n

p−2 . p−1

(4.2)

Next, consider an arbitrary polynomial P (x) = a0 + a1 x + · · · + an xn with integral coefficients. For any integer k ≥ 0, let ωk (P ) = inf{vp (aj ) | j ≥ k} . Clearly ω0 (P ) ≤ ω1 (P ) ≤ · · · ≤ ωk (P ) ≤ · · · and ωk (P ) = ∞ for k > n . Observe also that vp (P (t)) ≥ inf{a0 , a1 t, . . . , an tn } for any integer t ∈ Z, and consequently vp (P (t)) = inf{vp (a0 ), vp (a1 ), . . . , vp (an )} ≥ ω0 (P ) .

(4.3)

Lemma 4.4 Let P and Q be two polynomials with rational coefficients such that P (x) = (x − t)Q(x) for some t ∈ Z. Then for all k ∈ N ωk+1 (P ) ≤ ωk (Q) . Proof. Set Q(x) = a0 + a1 x + · · · + an xn ,

P (x) = b0 + b1 x + · · · + bn+1 xn+1 .

Then bj+1 = aj − taj+1 for 0 ≤ j ≤ n − 1, bn+1 = an , whence for j = 0, . . . , n, aj = bj+1 + tbj+2 + · · · + tn−j bn+1 . This shows that vp (aj ) ≥ ωj+1 (P ) for any j ∈ N. Thus, given any k ∈ N, one has for j ≥ k vp (aj ) ≥ ωj+1 (P ) ≥ ωk+1 (P ) and consequently ωk (Q) ≥ ωk+1 (P ) . 

4. A Theorem of Skolem, Mahler, Lech

65

Corollary 4.5 Let Q be a polynomial with rational coefficients, let t1 , t2 , . . . , tk ∈ Z, and let P (x) = (x − t1 )(x − t2 ) · · · (x − tk )Q(x) . Then ωk (P ) ≤ ω0 (Q) . The main argument is the following lemma. Lemma 4.6 Let (dn )n∈N be any sequence of integers and let (bn )n∈N be the sequence defined by bn =

n   X n i p di . i i=0

where p is an odd prime number. If bn = 0 for infinitely many indices n, then the sequence (bn )n∈N vanishes. Proof. For n ∈ N, let Rn (x) =

n X

di pi

i=0

x(x − 1) · · · (x − i + 1) . i!

Then for t ∈ N, n   X t i p di Rn (t) = i i=0

 t and since = 0 for i > t, it follows that i bt = Rt (t) = Rn (t) (n ≥ t) .

(4.4)

Next, we show that for all k, n ≥ 0, ωk (Rn ) ≥ k

p−2 . p−1

For this, let Rn (x) =

n X

(n)

ck xk .

k=0

pi (n) Each ck xk is a linear combination, with integral coefficients, of numbers di , i! for indices i with k ≤ i ≤ n. Consequently,   i  p (n) . vp (ck ) ≥ inf vp di k≤i≤n i!

66

Chapter IV. Rational Series in One Variable

In view of Eq. (4.2), this implies   p−2 p−2 (n) vp (ck ) ≥ inf i ≥k p−1 p−1 which in turn shows that ωk (Rn ) ≥ k

p−2 . p−1

(4.5)

Consider now any coefficient bt of the sequence (bn )n∈N . We shall see that vp (bt ) ≥ k

p−2 p−1

for any integer k, which of course shows that bt = 0. For this, let t1 < t2 < · · · < tk be the first k indices with bt1 = · · · = btk = 0, and let n ≥ sup(t, tk ). By Eq. (4.4), Rn (ti ) = bti = 0 for i = 1, . . . , k. Thus Rn (x) = (x − t1 )(x − t2 ) · · · (x − tk )Q(x)

(4.6)

for some polynomial Q(x) with integral coefficients. By Corollary 4.5, one has ωk (Rn ) ≤ ω0 (Q) .

(4.7)

Next, by Eq. (4.4), vp (bt ) = vp (Rn (t)) and by Eqs. (4.6), (4.3) and (4.7), vp (Rn (t)) ≥ vp (Q(t)) ≥ ω0 (Q) ≥ ωk (Rn ) . Thus, in view of Eq. (4.5), vp (bt ) ≥ k

p−2 p−1

for all k ≥ 0.  P Lemma 4.7 Let S = an xn ∈ Z[[x]] be a regular rational series and let (λ, µ, γ) be a linear representation of S of dimension k with integral coefficients. For any odd prime p not dividing det(µ(x)), the annihilator ann(S) is quasi2 periodic of period at most pk . Proof. Let p be an odd prime that does not divide det(µ(x)). Let n 7→ n be the canonical morphism from Z onto Z/pZ. Since det(µ(x)) = det(µ(x)) 6= 0, 2 the matrix µ(x) is invertible in Z/pZ, and there is an integer N ≤ pk with µ(xN ) = I . Reverting to the original matrix, this means that µ(xN ) = I + pM for some matrix M with integral coefficients.

67

4. A Theorem of Skolem, Mahler, Lech Consider now a fixed integer j ∈ {0, . . . , N − 1} and set for n ≥ 0 bn = aj+nN . Then j+nN

bn = λµ(x

n   X n i p λµ(xj )M i γ . )γ = λµ(x )(I + pM ) γ = i i=0 j

n

Thus, setting di = λµ(xj )M i γ, one obtains n   X n i p di . bn = i i=0

In view of Lemma 4.6, the sequence (bn )n≥0 either vanishes or contains only finitely many vanishing terms. Thus, the annihilator of S is quasi-periodic with 2 period less than pk .  Proof of Proposition 4.3. Let (λ, µ, γ) be a regular linear representation of S, and let q be a common multiple of the denominators of the coefficients in λ, µ and γ. Then (qλ, qµ, qγ) is a linear representation of the regular series P n+2 S′ = q an xn . Clearly ann(S) = ann(S ′ ). By Lemma 4.7, the set ann(S ′ ) is quasi-periodic. Thus ann(S) is quasi-periodic.  We now turn to the second part of the proof. For this, we consider the ring Z[y1 , . . . , ym ] of polynomials over Z in commutative variables y1 , . . . , ym and the quotient field Q(y1 , . . . , ym ) of rational functions. An element in either one of these sets will be denoted indistinctly without or with an enumeration of the variables. As usual, if P ∈ Q(y1 , . . . , ym ) and a1 , . . . , am ∈ Q, then P (a1 , . . . , am ) is the value of P at that point. The result to be proved is the following. P Proposition 4.8 Let S = an xn be a regular rational series with coefficients in the field Q(y1 , . . . , ym ). Then ann(S) is quasi-periodic. We start with the following well-known property of polynomials. Lemma 4.9 Let K be a (commutative) field, and let P ∈ K[y1 , . . . , ym ]. Let δi be the degree of P in the variable yi . Assume that there exist subsets A1 , . . . , Am of K with Card(Ai ) > δi for i = 1, . . . , m such that P (a1 , . . . , am ) = 0 for all (a1 , . . . , am ) ∈ A1 × · · · × Am . Then P = 0.  P Corollary 4.10 Let S = an xn be any series with coefficients in K[y1 , . . . , ym ] and let H1 , . . . , Hm be arbitrary infinite subsets of K. For each (h1 , . . . , hm ) ∈ K m , let X Sh1 ,...,hm = an (h1 , . . . , hm )xn . Then

ann(S) =

\

(h1 ,...,hm )∈H1 ×···×Hm

ann(Sh1 ,...,hm ) .

68

Chapter IV. Rational Series in One Variable

Proof. It follows immediately from Lemma 4.9 that an = 0 iff an (h1 , . . . , hm ) = 0 for all (h1 , . . . , hm ) ∈ H1 × · · · × Hm .  Lemma 4.11 Let P ∈ Z[y1 , . . . , ym ], P 6= 0. For all but a finite number of prime numbers p, there exists a subset H ⊂ Zm of the form H = (k1 , . . . , km ) + pZm

(4.8)

such that for all (h1 , . . . , hm ) ∈ H, P (h1 , . . . , hm ) 6≡ 0 mod p . Proof. Let P =

X

im ci1 ,i2 ,...,im y1i1 y2i2 · · · ym .

Let δi be the degree of P in the variable yi , and let p be any prime number strictly greater than the δi ’s and not dividing all the coefficients ci1 ,i2 ,...,im . Again let n 7→ n be the morphism from Z onto Z/pZ. The polynomial X im P = ci1 ,i2 ,...,im y1i1 y2i2 · · · ym

is a non vanishing polynomial with coefficients in Z/pZ. Since p > δi for i = 1, . . . , m, it follows from Lemma 4.9 that there exists (k1 , . . . , km ) ∈ Zm such that P (k 1 , . . . , km ) 6= 0. This proves the lemma.  Proof of Proposition 4.8. Let (λ, µ, γ) be a linear representation of S of dimension k. As in the proof of Proposition 4.3, consider a common multiple q ∈ Z[y1 , . . . , ym ] of the denominators of the coefficients λ, µ and γ. Pofn+2 Then (qλ, qµ, qγ) is a linear representation of the series S ′ = q an xn and ′ ann(S ) = ann(S). Thus we may suppose that the coefficients of λ, µ and γ are in Z[y1 , . . . , ym ]. Let P = det(µ(x)) ∈ Z[y1 , . . . , ym ]. Since S is regular, P 6= 0 and by Lemma 4.11, there exists a prime number p and an infinite H ⊂ Zn of the form Eq. (4.8) such that
det µ(x)(h1 , . . . , hm ) 6≡ 0 mod p for all (h1 , . . . , hm ) ∈ H. Setting X Sh1 ,...,hm = an (h1 , . . . , hm )xn n

this implies, in view of Lemma 4.7, that for all (h1 , . . . , hm ) ∈ H, the set 2 2 ann(Sh1 ,...,hm ) is quasi-periodic with a period at most pk . Thus r = (pk )! is a period for all these annihilators. In view of Lemma 4.2, the set \ ann(Sh1 ,...,hm ) (h1 ,...,hm )∈H

is quasi-periodic. By Corollary 4.10, this intersection is the set ann(S). Thus the proof is complete.  It is convenient to introduce the following

4. A Theorem of Skolem, Mahler, Lech

69

Definition A (commutative) field K is a SML field (Skolem-Mahler-Lech field) if K satisfies Theorem 4.1. We have seen already that the field Q of rational numbers, and the field Q(y1 , . . . , ym ) are SML fields. Proposition 4.12 Let K and L be fields. If L is an SML field and K is a finite algebraic extension of L, then K is an SML field. P Proof. Let S = an xn be a rational series over K. Let k be the dimension of K over L, and let φ1 , . . . , φk be L-linear functions K → L such that, for any h∈K h = 0 ⇐⇒ φi (h) = 0, ∀ i = 1, . . . , k . Define Si =

X n

φi (an )xn ∈ L[[x] .

Then, by the choice of the function φi , one has \ ann(S) = ann(Si ) .

(4.9)

1≤i≤k

Thus, it suffices, by Lemma 4.2 to prove that the series Si are rational over L. By Proposition I.5.1, there exists a finite dimensional subvector space M of K[[x]], containing S and which is stable, that is closed for the operation T 7→ T ◦ x. Since K has finite dimension over L, the space M also has finite dimension over L. The functions φi , extended to series φi : K[[x]] → L[[x]] by φi

X n

 X φi (bn )xn bn xn = n

are L-linear. Consequently, φi (M ) is a finite dimensional vector space over L. Since φi (T ◦ x) = φi (T ) ◦ x, the space φi (M ) is stable. Moreover, it contains the series Si = φi (S). Thus, again by Proposition I.5.1, each series Si is rational over L.  Proof of Theorem 4.1. Let S be a rational series with coefficients in K. Then by Proposition 1.4, there is a polynomial P such that S − P is regular. Since ann(S − P ) and ann(S) differ only by a finite set, it suffices to prove the result for S − P . Thus we may assume that S is regular. Let (λ, µ, γ) be a linear representation of S, and let K ′ be the subfield of K over Q generated by the set Z of coefficients of λ, µ(x), γ. Then S has coefficients in K ′ and we may assume that K is a finite extension of Q, that is K = Q(Z) for a finite set Z. Let Y be a maximal subset of Z that is algebraically independent over Q. The field Q(Y ) is isomorphic to the field Q(y1 , . . . , ym ) with Y = {y1 , . . . , ym }. In view of Proposition 4.8, the field Q(Y ) is a SML field. Next, K is a finite algebraic extension of Q(Y ). By Proposition 4.12, the field K is a SML field. This concludes the proof. 

70

Chapter IV. Rational Series in One Variable

Exercises for Chapter IV P P∞ n 4.1 Set B(x) = ∞ d xn with integers bn , dn related n=0 bn x , D(x) = n=0 P∞n pn xn as in Lemma 4.6. Show that B(x) = n=0 dn (1−x) n+1 .

Notes to Chapter IV The notion of an exponential polynomial is a classical one. The formalism we use here is from Reutenauer (1982). It allows to give an algebraic proof of Benzaghou’s theorem. His proof was based on analytic techniques. The algebraic method makes it possible to prove Benzaghou’s theorem in characteristic p. Some modifications are necessary, since in that case, the exponential polynomial may not exist nor be unique. P´ olya’s theorem is extended to general fields by B´ezivin (1984). There are a great number of arithmetic and combinatorial properties of linear recurrence sequences. The use of symmetric functions to derive divisibility properties is illustrated by Dubou´e (1983). Lascoux (1986) gives numerous applications of expressions of the exponential polynomial by means of symmetric functions. For a rich collection of formulas and results about symmetric functions, see Lascoux and Sch¨ utzenberger (1985). The proof of the Skolem-Mahler-Lech theorem given here is due to Hansel (1986). The original proofs, by Skolem (1934), Mahler (1935), and Lech (1953) depend on p-adic analysis. An openP problem, stated by C. Pisot, is the following. Is it decidable, for a rational series an xn , whether there exists an n such that an = 0? It is decidable whether there exist infinitely many n with an = 0 (Berstel and Mignotte 1976). An extension of P´ olya’s theorem to several noncommutative variables is studied in Reutenauer (1980b). For this, define as follows the unambiguous rational operations on languages (see Eilenberg and Sch¨ utzenberger 1969): The union L1 ∪ L2 is unambiguous if the sets are disjoint. The product L1 L2 is unambiguous if u, u′ ∈ L1 , v, v ′ ∈ L2 , and uv = u′ v ′ imply u = u′ , v = v ′ . The star operation L 7→ L∗ is unambiguous if L is the basis of a free submonoid of A∗ . It can be shown that any rational language is obtained from finite languages by unambiguous operations (Eilenberg 1974). The unambiguous rational operations on series are defined as follows. A rational operation (sum, product, star) on series is unambiguous if the corresponding operation on the support (union, product, star) is unambiguous. A rational series S ∈ QhhAii is unambiguous if it is obtained from polynomials using only unambiguous rational operations. It is not very difficult to see that any unambiguous rational series has only finitely many prime factors. The problem is the converse property, which we state as a conjecture. Conjecture If a rational series has only finitely many prime factors, then it is unambiguous. According to P´ olya’s theorem, this holds for one variable. Partial results are proved in Reutenauer (1980b).

Chapter V

Changing the Semiring If K is a subsemiring of L, each K-rational series is clearly L rational. The main problem considered in this chapter is the converse: how to determine which of the L-rational series are rational over K. This leads to the study of semirings of a special type, and also shows the existence of remarkable families of rational series. In the first section, we examine principal rings from this aspect. Fatou’s Lemma is proved and the rings satisfying this lemma are characterized. Section 2 contains several results on rational series with nonnegative coefficients. The main result (Theorem 2.10) is a characterization of R+ -rational series in one variable. In the last section, Fatou extensions are introduced. We show in particular that Q+ is a Fatou extension of N (Theorem 3.3).

1

Rational Series over a Principal Ring

Let K be a commutative principal ring and let F be its quotient field. Let S ∈ KhhAii be a formal series over A with coefficients in K. If S is a rational series over F , is it also rational over K? This question admits a positive answer, and there is even a stronger result, namely that S has a linear representation of minimal dimension (that is, equal to its rank) with coefficients in K. Theorem 1.1 (Fliess 1974a) Let S ∈ KhhAii be a series which is rational of rank n over F . Then S is rational over K and has a linear representation over K of dimension n. Proof. Let (λ, µ, γ) be a reduced linear representation of S over F . According to Corollary II.2.3, there exist polynomials P1 , . . . , Pn , Q1 , . . . , Qn ∈ F hAi such that for w ∈ A∗ µw = ((S, Pi wQj ))1≤i,j≤n . Let d be an element in K \ 0 such that dPi , dQj ∈ KhAi and dλ ∈ K 1×n . Then for any polynomial P ∈ KhAi d3 λµP = (dλ)((S, dPi P dQj ))i,j ∈ K 1×n , 71

72

Chapter V. Changing the Semiring

since (S, R) ∈ K whenever R ∈ KhAi. Consequently, λµ(KhAi) ⊂

1 1×n K . d3

This shows that λµ(KhAi), considered as a submodule of a free K-module of rank n, is also free and has rank ≤ n. It suffices now to apply Lemma II.1.2.  This theorem admits the following corollary, known as Fatou’s Lemma. Corollary 1.2 (Fatou 1904) Let P (x)/Q(x) ∈ Q(x) be an irreducible rational function such that the constant term of Q is 1. If the coefficients of its series expansion are integers, then P and Q have integral coefficients. P Proof. We have Q(0) = 1. Then S = an xn = P (x)/Q(x) is a rational series. Let (λ, µ, γ) be a reduced linear representation of S. Since Z is principal, this representation is similar, by Theorem 1.1 and Theorem II.2.4, to a representation over Z. In particular, the characteristic polynomial of µ(x) has integral coefficients. Now, Q(x) is the reciprocal polynomial of this polynomial (Proposition IV.1.2). Thus Q(x) has integral coefficients, and so does P = SQ.  The previous result holds for rings other than the ring Z of integers. We shall characterize these rings completely. Let K be a commutative integral domain and let F be its quotient field. Let M be an F -algebra. An element m ∈ M is quasi-integral over K if there exists an injection of the K-module K[m] into a finitely generated K-module. Proposition 1.3 If m ∈ M is quasi-integral over K, then there exists a finitely generated K-submodule of M containing K[m]. Proof. There exists a finitely generated K-module N and a K-linear injection K[m] → N . Since K[m] is contained in some F -algebra, it is torsion-free over K. Thus the injection extends to an F -linear injection i : F [m] → N ⊗K F . Consequently F [m] has finite dimension over K and m is algebraic over F . Let p : N ⊗ F → i(F [m]) be an F -linear projection. Then p(N ) = p(N ⊗ 1) is a finitely generated K-module containing i(K[m]) and contained in i(F [m]). Consequently, its inverse image by i, say N1 , is a finitely generated K-module and K[m] ⊂ N1 ⊂ F [m] ⊂ M .  Corollary 1.4 An element m ∈ F is quasi-integral over K if and only if there exists d ∈ K \ 0 such that dmn ∈ K for all n ∈ N. P Proof. Indeed, K[m] is the set of all expressions ni=0 αi mi , with αi ∈ K. 

Corollary 1.5 If M is a commutative algebra, then the set of elements of M which are quasi-integral over K is a subring of M . 

1. Rational Series over a Principal Ring

73

Definition The domain K is called completely integrally closed if any m in F which is quasi-integral over K is already in K. Observe that an element in F which is integral over K is also quasi-integral over K. Thus, if K is completely integrally closed, it is integrally closed. Theorem 1.6 (Chabert 1972) The following conditions are equivalent. (i) The domain K is completely integrally closed. (ii) For any irreducible rational function P (x)/Q(x) ∈ F (x) whose series expansion has coefficients in K, and such that the constant term of Q is 1, both P and Q have coefficients in K. We use the following lemma Lemma 1.7 Let m be a matrix in F n×n which is quasi-integral over K. Then the coefficients of the characteristic polynomial of m are quasi-integral over K. Proof. Let P (t) = tn + a1 tn−1 + · · · + an ∈ F [t] be the characteristic polynomial of m. Since m is quasi-integral over K, there exists, by Proposition 1.3, a finitely generated K-submodule of F n×n containing all powers of m. Thus there exists some d ∈ K \ 0 such that dmk ∈ K n×n for all k ∈ N. Consequently, since ±ai is a sum of products of i entries of m, da1 , d2 a2 . . . , dn an ∈ K . Let λ be an eigenvalue of m. Then dλ is integral over K. Indeed, 0 = dn P (λ) = (dλ)n + da1 (dλ)n−1 + · · · + dn an . Consequently, the K-algebra L = K[dλ] is a finitely generated K-module. The element λ is in the quotient field E of L, and there exists q ∈ GLn (E) such that   λ ∗ ··· ∗  0 ∗ · · · ∗   m′ = q −1 mq =  . ..   .. . 0 ∗ ··· ∗

Let d′ be a common denominator of the coefficients of q and q −1 , that is such that d′ q and d′ q −1 have coefficients in L. Then for all k ∈ N (d′2 d)m′k = (d′ q −1 )dmk (d′ q) ∈ Ln×n .

Thus (d′2 d)λk ∈ L, whence K[λ] ⊂ (d′2 d)−1 L. This shows that λ is quasiintegral over K. Since all eigenvalues of m are quasi-integral, the same holds for the coefficients ai by Corollary 1.5.  Proof of Theorem 1.6. Assume that K is completely integrally closed. Let P (x)/Q(x) be a function satisfying the hypotheses of (ii). We have Q(0) = 1. The series X S= an xn = P (x)/Q(x)

74

Chapter V. Changing the Semiring

is F -rational and has coefficients in K. Let (λ, µ, γ) be a reduced linear representation of S. By Corollary II.2.3, the matrix µ(x) is quasi-integral over K. In view of Lemma 1.7, the characteristic polynomial of µ(x) has coefficients in K, and since Q is its reciprocal polynomial (Proposition IV.1.2), the polynomial Q has coefficients in K, and the same holds for P = SQ. Assume conversely that (ii) holds. Let m ∈ F be quasi-integral over K. Then there exists d ∈ K \ 0 such that dmn ∈ K for all n ∈ N. Set P (x) = d, Q(x) = 1 − mx. Then X P (x)/Q(x) = d mn xn ∈ K[[x]] .

Thus by hypothesis Q(x) ∈ K[x], whence m ∈ K. This shows that K is completely integrally closed. 

2

Positive Rational Series

In this section, we study series with nonnegative coefficients. We start with the following result. Theorem 2.1 Sch¨ utzenberger (1970) If S ∈ NhhAii is an N-rational series, then S − supp(S) ∈ NhhAii is N-rational. Recall that L is the characteristic series of the language L. Proof (Salomaa and Soittola 1978). In view of Proposition I.5.1, there exist rational series S1 , . . . , Sn such that the N-submodule of NhhAii they generate is stable and contains S. By Lemma III.1.4, the supports supp(S1 ), . . . , supp(Sn ) are rational languages. Let L be the family of languages obtained by taking all intersections of supp(S1 ), . . . , supp(Sn ). Then L is a finite set of rational languages. The set L′ = {u−1 L | u ∈ A∗ , L ∈ L} is also a finite set of rational languages (Corollary III.1.6). Let T be the set of characteristic series of the languages in L′ . Let M be the finitely generated N-submodule of NhhAii generated by T and by the series Si′ = Si − supp(Si ) P for i = 1, . . . , n. We claim that if aj ∈ N and T = aj Sj , then T − supp(T ) is in M . P P Indeed, Sj = Sj′ +supp(Sj ), thus T = aj Sj′ +U , where U = aj supp(Sj ). Note P that supp(Sj′ ) ⊂ supp(Sj ), hence supp(T ) = supp(U ). We may write U = bk Tk where each integer bk is ≥ 1 and the Tk ∈ T have disjoint supports. This is done by keeping only the j’s with aj ≥ 1 and P by making the necessary intersections of supports. Hence U − supp(U ) = (bk − 1)Tk ∈ M and T − P supp(T ) = aj Sj′ + U − supp(U ) ∈ M .

2. Positive Rational Series

75

Since S is an N-linear combination of the Sj , S − supp(S) is in M by the claim. We show that M is stable, which will end the proof by Proposition I.5.1. Indeed, let u ∈ A∗ . Then u−1 T ∈ T by construction, hence in M , for any T in T. Consider u−1 Si′ = u−1 Si − supp(u−1 Si ). Since u−1 Si is an N-linear combination of the Sj , we deduce that u−1 Sj′ is in M .  We now consider series of the form X an xn

with all coefficients in R+ . If such a series is the expansion of a rational function, it does not imply in general that it is R+ -rational (see Exercise 2.2). We shall characterize those rational functions over R whose series expansion is R+ rational. We call them R+ -rational functions. Theorem 2.2 (Berstel 1971) Let f (x) be an R+ -rational function which is not a polynomial, and let ρ be the minimum of the moduli of its poles. Then ρ is a pole of f , and any pole of f of modulus ρ has the form ρθ, where θ is a root of unity. Observe that the minimum of the moduli of the poles of a rational function is just the radius of convergence of the associated series. We start with a lemma. Lemma 2.3 Let f (x) bePa rational function which is not a polynomial and with a series expansion an xn having nonnegative coefficients. Let ρ be the minimum of the moduli of the poles of f . Then ρ is a pole of f , and the multiplicity of any pole of f of modulus ρ is at most that of ρ. Proof. Let z ∈ C, |z| < ρ. Then X X |f (z)| = an z n ≤ an |z|n = f (|z|) .

(2.1)

Let z0 be a pole of modulus ρ, and let π be its multiplicity. Assume that the multiplicity of ρ as a pole of f is less than π. Then the function g(z) = (ρ − z)π f (z) is analytic in the neighborhood of ρ, and g(ρ) = 0, whence lim (ρ − ρr)π f (ρr) = 0 .

r→1,r 1 and |λ0 λi | < 1 for i = 1, . . . , r. P Lemma 2.7 Let S = an xn be a rational series over R with nonnegative coefficients which is not a polynomial and which has a dominating eigenvalue. P an n Then for some real number s > 0, the series x has a strictly dominating sn eigenvalue. Proof. Since S has a dominating eigenvalue, one has for large enough n, an = P (n)λn +

r X

Pi (n)λni

i=1

with |λ| > |λi | for i = 1, . . . , r and polynomials P, P1 , . . . , Pr . If α is the leading coefficient of P , then for n → ∞ an ∼ αnd λn with d = deg(P ). Thus α > 0 and λ is real, λ > 0. Set λ0 = λ. Let s′ > 0 be a real number with λ > s′ > |λi | for i = 1, . . . , r, and set λ′i = λi /s′ . Then of course λ′0 > 1 > |λ′i |. Next, let s′′ be such that 1 < s′′ < λ′0 < s′′2 . Setting λ′′i = λ′i /s′′ , it follows that |λ′′0 λ′′i | =

λ′0 |λ′0 λ′i | < 1. Thus the number s = s′ s′′ gives the desired result.  P Lemma 2.8 Let S = an xn ∈ R+ [[x]] be a rational series having a strictly dominating eigenvalue. Then for large enough n an+1 > an

(2.6)

79

2. Positive Rational Series Proof. For large enough n, one has an = P (n)λn +

r X

Pi (n)λni

(2.7)

i=1

with λ ∈ R, λ > 1 and |λλi | < 1, hence also |λi | < 1/λ < 1 < λ. This shows that an ∼ αnd λn . n→∞

where d is the degree of P and α is the coefficient of td in P (t). This implies that an+1 ∼ λ. an n→∞ Since λ > 1 and an ∈ R+ , this shows that an+1 > an for large enough n.  Lemma 2.9 Let U be a rational series over R having a strictly dominating eigenvalue with multiplicity m. Then there exists an integer p suchP that U is the merge of p rational series having the following property: if S = an xn is any one of them, then (i) S has a dominating eigenvalue λ with multiplicity m. (ii) There exists a polynomial R = 1 − α1 x − · · · − αk xk ∈ R[[x]] which has the simple root 1/λ and such that α1 > 1, α1 + α2 > 1, . . . , α1 + · · · + αk > 1 kαk 2α2 α1 + ···+ k > 0 + λ λ2 λ

(2.8)

Moreover, if m = 1, then R is a denominator for S. Proof. 1. Let λ0 , . . . , λr be the distinct nonzero eigenvalues of U . Then for large enough n X un = λni Pi (n) , 0≤i≤r

where U =

P

un xn , Pi (t) ∈ C[[t]] \ 0 and, by hypothesis

λ0 ∈ R, λ0 > 1,

λ0 |λi | < 1 for i = 1, . . . , r .

2. Define a polynomial R as follows: if λ0 is a simple eigenvalue, then R is the minimal denominator of U , otherwise let R be a divisor of the minimal denominator of U in R[x] such that R has the simple root 1/λ0 . (p) (p) 3. Let R(p) = 1 − α1 x − · · · − αk xk be the polynomial whose roots are the pth powers of those of R, with the same multiplicities. We show that for some (p) p, we have Eq. (2.8) (with αi replaced by αi ). Let λ0 = µ0 , . . . , µk be the inverses of the roots of R. Then, by assumption µ0 > 1 > µ0 |µi | for i = 1, . . . , k .

80

Chapter V. Changing the Semiring (p)

Let σi

be the ith elementary symmetric function of the µpi ’s. Then (p)

σ1

= µp0 + · · · + µpk > µp0 − k

(p)

since σ1

∈ R and |µi | < 1 for i = 1, . . . , k. Moreover, for i = 2, . . . , k   X k+1 (p) (p) (p) (p) σi = µj1 µj2 · · · µji < i 0≤j1 1 (p) (p) (p) (p) and σ1 ± σi > 1 for i = 2, . . . , k. Now, observe that α1 = σ1 and (p) (p) αi = ±σi for i = 2, . . . , k. Hence (p)

(p)

(p)

(p)

(p)

α1 > 1, α1 + α2 > 1, . . . , α1 + · · · + αk > 1 . (p)

(p)

Moreover, since α1 = σ1 is equivalent to µp0 when p → ∞, µ0 > 1 and the (p) αi (i = 1, . . . , k) are bounded, we obtain for large enough p (p)

(p)

(p)

α α1 α + 2 22p + · · · + k kkp > 0 . µp0 µ0 µ0 4. Let p be as in step 3, and define Sj by X Sj = uj+np xn . n

Then U is the merge of S0 , . . . , Sp−1 . We have for large enough n X p uj+np = (λi )n λji Pi (j + np) .

(2.9)

0≤i≤r

This shows that the eigenvalues of Sj are among λp0 , . . . , λpr (by unicity of the exponential polynomial, see Corollary IV.2.2). Moreover, λp0 6= λpi for i = 1, . . . , k, hence λp0 is a eigenvalue of Sj , and hence Sj has the dominating eigenvalue λp0 . Since the polynomials P0 (t) and P0 (j + tp) have the same degree, the multiplicity of λp0 for Sj is m = deg(P0 ) + 1. Now R(p) has the simple root 1/µp0 = 1/λp0 with multiplicity 1 (because λp0 6= λpi if i 6= 0) and the relations (2.8) hold. If m = 1, then R was chosen to be the minimal denominator of U , hence R(p) is a multiple of the minimal denominator of Sj . Indeed, by Eq. (2.9) and Corollary IV.2.2, each of Sj is one of the λpi with multiplicity Q eigenvalue ≤ deg(Pi ). Now, R(x) = (1−λi x)deg(Pi )+1 (see Section IV.2), hence R(p) (x) = i Q (1 − λpi x)deg(Pi )+1 by definition.  i

Proof of Theorem 2.4. 1. We argue by induction on the multiplicity m of the dominating eigenvalue λ of S. By Lemma 2.7 we may assume that S has a strictly dominating eigenvalue. Hence, by Lemma 2.8 and Lemma 2.5(i), we may assume that an+1 > an

2. Positive Rational Series

81

for any n. Now, by Lemma 2.9 and Lemma 2.5(ii), we may further assume that there exists a polynomial R = 1 − α1 x − · · · − αk xk having 1/λ as a simple root, satisfying Eq. (2.8), and which is a denominator of S if m = 1. P P T 2. Let S = = , with P, Q ∈ R[x], T ∈ R[[x]]. The series T = bn xn Q R is of course rational, and with the notations of Lemma 2.6, we obtain (λ0 being dominating)   α1 αk bn ∼ dα + · · · + k k λn0 . n→∞ λ0 λ0 Since an ≥ 0, we must have α ≥ 0, hence bn ≥ 0 for large n. P n Let T [h] = n>h bn x . If m = 1, then R is a denominator of S, hence [h] T = 0 for large h. If m ≥ 2, then, as 1/λ0 is a simple root of R, λ0 is a eigenvalue of T and the other eigenvalues are among those of S. Hence T has a dominating eigenvalue of multiplicity m − 1. The same holds for T [h] by Lemma 2.5(i). Thus, in both cases, T [h] is R+ -rational by induction, for some h. 3. Before going into the technicalities of the proof, we consider, from a heuristic point of view, the case where h = 0, that is where bn ≥ 0 for any n. 1 Then T is R+ -rational by induction and S = T . Thus we only have to show R P 1 1 1 1 (where x∗ = xn = that is R+ -rational. Now = x∗ ) and ∗ R R Rx 1−x Rx∗ = (1 − α1 x − · · · − αk xk )(1 + x + x2 · · · + xn + · · · )

= 1 + (1 − α1 )x + (1 − α1 − α2 )x2 + · · · + (1 − α1 − · · · − αk−1 )xk−1 + (1 − α1 − · · · − αk )(xk + xk+1 + · · · + xn + · · · )

= 1 − γ1 x − · · · − γk−1 xk−1 − γk xk x∗ , where γ1 = α1 − 1

γ2 = α2 + α1 − 1 .. . γk = αk + · · · + α1 − 1 . 1 = (γ1 x + · · · + γk−1 xk−1 + By Eq. (2.8), the γ’s are nonnegative, so that Rx∗ 1 γk xk x∗ )∗ is R+ -rational. Hence is also R+ -rational, which concludes the R proof in this special case. 4. There remains the problem of treating the case h > 0. We may suppose h ≥ k. We show that for some polynomial Rh with nonnegative coefficients, one has   γk ah xh+k xh+1 1 T [h] + (2.10) S [h] = ah + + Rh , 1−x R 1−x

82

Chapter V. Changing the Semiring

where the γ’s are defined in step 3. Note that T [h] and steps 2 and 3), so S [h] is R+ -rational. Thus

1 are R+ -rational (by R

S = a0 + · · · + ah xh + S [h] is also R+ -rational. P cn xn with 5. Let C = n≥0

c0 = a 0 ,

cn = an − an−1 (n ≥ 1) .

Of course C = (1 − x)S and by 1. C has nonnegative coefficients. We shall see that   1−x xh+k C [h] = T [h] + γk ah + Rh . R 1−x

(2.11)

Since, as is easily verified, S [h] =

1 (C [h] + ah xh+1 ) , 1−x

the formula (2.10) follows from Eq. (2.11). 6. Since T = SR,we have for n > h (hence for n ≥ k, because h ≥ k) bn = an − α1 an−1 − · · · − αk−1 an−k+1 − αk an−k

= an − (1 + γ1 )an−1 − · · · − (γk−1 − γk−2 )an−k+1 − (γk − γk−1 )an−k

= an − an−1 − γ1 (an−1 − an−2 ) − · · · −γk−1 (an−k+1 − an−k ) − γk an−k

= cn − γ1 cn−1 − · · · − γk−1 cn−k+1 − γk an−k .

This relation can be developed further, when n ≥ h + k, since ap = cp + ap−1 . We obtain bn = cn − γ1 cn−1 − · · · − γk−1 cn−k+1 − γk cn−k − γk cn−k−1 − · · · − γk ch+1 − γk ah . Summing up these relations, we get X  X n n bn x = cn x (1 − γ1 x − · · · − γk xk − γk xk+1 − · · · ) n>h

n>h

−

γk ah xh+k − Rh , 1−x

with Rh = (γ1 ch + γ2 ch−1 + · · · + γk−1 ch−k+2 + γk ah−k+1 )xh+1 + (γ2 ch + · · · + γk−1 ch−k+3 + γk ah−k+2 )xh+2

+ · · · + (γk−1 ch + γk ah−1 )xh+k−1 .

83

3. Fatou Extensions This polynomial has nonnegative coefficients. We have T [h] = C [h] 1 − γ1 x − · · · − γk−1 xk−1 − Formula (2.11) follows because

γk xk
ah xh+k − γk − Rh . 1−x 1−x

1−x xk
−1 = 1 − γ1 x − · · · − γk−1 xk−1 − γk R 1−x

(see step 3). The proof of the theorem is complete. 

The two preceding theorems give the following characterization of R+ -rational series. Theorem 2.10 A series S ∈ R+ [[x]] is R+ -rational if and only if it is the merge of rational series having a dominating eigenvalue. Proof. Let S be a R+ -rational series and let m be the number of its eigenvalues. We argue by induction on m, the case m = 1 being clear. If m > 1, then S is not a polynomial, since the only eigenvalue of a polynomial is 0. Since the nonvanishing eigenvalues of S are the inverses of the poles of the associated function (see Section IV.2), there exists, by Theorem 2.2, a real number ρ > 0 such that any eigenvalue of S with maximal modulus has the form ρθ, where θ is a root of unity. Let p be a common order of all these roots of unity and let S0 , S1 , . . . , Sp−1 be the series whose merge is S. By Lemma 2.5, each Si is R+ -rational and their eigenvalues are λp , with λ a eigenvalue of S. If p = 1, S has a dominating eigenvalue and we are done. Otherwise, each Si has strictly fewer eigenvalues than S, and is R+ -rational. This concludes the proof of the direct part. For the converse, it suffices to use Theorem 2.4 and Lemma 2.5. 

3

Fatou Extensions

According to Fatou’s Lemma (Corollary 1.2) any rational series in Q[[x]] with integral coefficients is rational in Z[[x]]. The same result holds for an arbitrary alphabet A, by Theorem 1.1. This leads to the following definition. Definition Let K ⊂ L be two semirings. Then L is a Fatou extension of K if every L-rational series with coefficients in K is K-rational. Theorem 3.1 (Fliess 1974a) If K ⊂ L are commutative fields, then L is a Fatou extension of K. Proof. This follows immediately from the expression of rationality by means of the rank of the Hankel matrix (Theorem II.1.5).  Theorem 3.2 Let K be a commutative Noetherian integral domain, and let F be its quotient field. Then F is a Fatou extension of K.

84

Chapter V. Changing the Semiring

Proof. Let S be an F -rational series with coefficients in K and let (λ, µ, γ) be a reduced linear representation of S. According to Corollary II.2.3, there exists d ∈ K \ 0 such that dµA∗ ⊂ K n×n . Consequently, the algebra µ(KhAi) is contained in d−1 K n×n which is a Noetherian K-module. It follows that µ(KhAi) is a finitely generated K-module and this in turn shows that the syntactic algebra of S over K (which is equal to µ(KhAi)) is a finitely generated K-module. By Theorem II.1.1, the series S is rational.  Theorem 3.3 (Fliess 1975) The semiring Q+ is a Fatou extension of N. We need some preliminary lemmas. Lemma 3.4 (Eilenberg and Sch¨ utzenberger 1969) The intersection of two finitely generated submonoids of an Abelian group is still a finitely generated submonoid. Proof. Let M1 and M2 be two finitely generated submonoids of an Abelian group G, with law denoted by +. There exist integers k1 , k2 and surjective monoid morphisms φi : Nki → Mi , i = 1, 2. Let k = k1 + k2 and let S be the submonoid of Nk = Nk1 × Nk2 defined by S = {x = (x1 , x2 ) ∈ Nk | φ1 x1 = φ2 x2 } . Let p1 : Nk → Nk1 be the projection. Then M1 ∩ M2 = φ1 ◦ p1 (S) . Thus it suffices to prove that S is finitely generated. Observe that S satisfies the following condition x, x + y ∈ S =⇒ y ∈ S .

(3.1)

Indeed, since φ1 x1 = φ2 x2 and φ1 x1 + φ1 y1 = φ2 x2 + φ2 y2 and since all these elements are in G, it follows that φ1 y1 = φ2 y2 , whence y ∈ S. Let X be the set of minimal elements of S (for the natural ordering of Nk ). For all z ∈ S, there is x ∈ X such that x ≤ z. Thus z = x + y for some y ∈ Nk and by Eq. (3.1), y ∈ S. This shows by induction that X generates S. In view of the following well-known lemma, the set X is finite.  Lemma 3.5 Every infinite sequence in Nk contains an infinite increasing subsequence. Proof. By induction on k. Let (un ) be a sequence of elements of Nk . If k = 1, either the sequence is bounded, and one can extract a constant sequence, or it is unbounded, and one can extract an strictly increasing subsequence. For k > 1, one first extracts a sequence that is increasing in the first coordinate, and then uses induction for this subsequence. 

85

3. Fatou Extensions

Lemma 3.6 (Eilenberg and Sch¨ utzenberger 1969) Let I be a set and let M be a finitely generated submonoid of NI . Then the submonoid M ′ of NI given by M ′ = {x ∈ NI | ∃n ≥ 1, nx ∈ M } is finitely generated. Proof. Let x1 , . . . , xp be generators of M . Let C = {x ∈ NI | ∃λ1 , . . . , λp ∈ Q+ ∩ [0, 1] : x =

X

λi xi } .

Then C contains each xi and is a set of generators for M ′ . Indeed, if nx = P λi xi ∈ M for some n ≥ 1 and some λi ∈ N, then x=

Xj λi k n

xi +

X λi n

−

j λ k i

n

xi ,

where ⌊z⌋ is the integral part of z. Thus, it suffices to show that C is finite. Let E be the subvector space of RI generated by M ′ . Since E has finite dimension, there exists a finite subset J of I such that the R-linear function p J : E → RJ (pJ is the projection RI → RJ ) is injective. The image of C by pJ is contained in NJ , and it is also contained in the set X K = {y ∈ RJ | ∃λ1 , . . . , λp ∈ [0, 1] : y = λi yi } , where yi = pJ (xi ). Now K is compact and NJ is discrete and closed. Thus K ∩ NJ is finite. It follows that C is finite. 

Proof of Theorem 3.3. Let S be a Q+ -rational series with coefficients in N. We use systematically Proposition I.5.1. There exists a finitely generated stable Q+ -submodule in Q+ hhAii that contains S. Denote it by MQ+ . Similarly, the series S is Q-rational with coefficients in Z, and therefore S is Z-rational. Thus, there is a finitely generated Z-submodule in ZhhAii that contains S, say MZ . Then M = MQ+ ∩ MZ is a stable N-submodule of NhhAii containing S, and it suffices to show that M is finitely generated. Let T1 , . . . , Tr be series in MQ+ generating it as a Q+ -module, and let MQ′ + =

X

NTi .

This is a finitely generated N-module. Since MZ is also a finitely generated N-module, the N-module M ′ = MZ ∩ MQ′ + ⊂ NhhAii is finitely generated (this follows from Lemma 3.4, noting that N-module = commutative monoid). Consequently, M = {T ∈ NhhAii | ∃n ≥ 1, nT ∈ M ′ }

86

Chapter V. Changing the Semiring

is, in view of Lemma 3.6, a finitely generated N-module. Finally, the N-module M ∩ MZ is finitely generated by Lemma 3.4. Since M = M ∩ MZ , this proves the theorem.  We now give two examples of extensions which are not Fatou extensions. Example 3.1 The ring Z is not a Fatou extension of N. Consider the series X (|w|a − |w|b )2 w . S= w∈{a,b}∗

This series is Z-rational (it is the Hadamard square of the series considered in Example III.3.1) and has coefficients in N. However, it is not N-rational, since otherwise its support would be a rational language (Section III.1), and also the complement of its support. In Example III.3.1, it was shown that this set is not the support of any rational series. Example 3.2 The √ semiring R+ is not a Fatou extension of Q+ (Reutenauer 1977a). Let α = (1/ 5)/2 be the golden ration and let S be the series X (α2(|w|a −|w|b ) + α−2(|w|a −|w|b ) )w, . S= w∈{a,b}∗

Since S = (α2 a + α−2 b)∗ + (α−2 a + α2 b)∗ , the series S is R+ -rational. Moreover, since α is an algebraic integer over Z and 1/α is its conjugate, one has for all n∈N α2n + α−2n ∈ Z . Consequently, S has coefficients in N. Assume that S is Q+ -rational. Then by Theorem 3.3, it is N-rational. However, the language S −1 (2) = {w | (S, w) = 2} is S −1 (2) = {w ∈ {a, b}∗ | |w|a = |w|b } since x + 1/x > 2 for all x > 0, x 6= 1. Since the language S −1 (2) is not rational, the series S is not N-rational (Corollary III.2.6). Thus S is not Q+ -rational.

Exercises for Chapter V 2.1 a) Let P (x) = xk − α1 xk−1 − · · · − αk with αk 6= 0 be a polynomial with integral coefficients, and suppose that all its roots have modulus P ≤ 1. Show that these roots are roots of unity. (Consider the series S = an xn with an = λn1 + · · · + λnk

where the λi ’s are the roots of P with their multiplicities. The minimal polynomial Q of S has the same roots as P , but with multiplicity 1. The

87

3. Fatou Extensions

sequence (an ) takes only a finite number of distinct values, and thus Q has a multiple ofP the form xp − xq .) b) Let S = an xn be a rational series with coefficients in Z and with polynomial growth (that is there exist a constant C and an integer q such that |an | ≤ Cnq for all n ≥ 1). Show that there exist an integer p, and polynomials Pi ∈ Q[t] for 0 ≤ i ≤ p − 1, such that for all i ∈ {0, . . . , p − 1} ai+np = Pi (n) for all but a finite number of n. c) Show that if a polynomial P ∈ Q[t] satisfies P (n) ∈ Z

2.2

3.1 3.2

3.3 3.4

3.5

for n ∈ N ,

then   P is a linear combination with integral coefficients, of polynomials t(t − 1) · · · t − q + 1 t = . Show that if moreover P (n) ∈ N for all q q! P n ∈ N, then the series P (n)xn is N-rational. d) Use the previous questions to show that any Z-rational series in one variable with nonnegative coefficients and with polynomial growth is Nrational. (See P´ olya and Szeg¨ o 1964, Exercise 200 in Chapter 4 and Exercise 85 in Chapter 2.) P a) Let θ be a real number. Show that the series S = n≥0 (cos2 nθ)xn is a C-rational series. (Give an expression for S as a rational function by using the formula cos nθ = 1/2(einθ + e−inθ ).) b) Let 0 < a < c be integers and let θ be a real number with 0 < θ < π/2, such that cos θ = a/c. that the numbers cn cos nθ are integers. Show P Show 2n that the series T = (c cos2 nθ)xn is Z-rational with coefficients in N. c) Show that if c 6= a, then z = eiθ is not a root of unity (use the fact that z is an algebraic number of degree ≤ 2, and that the assumption that z is a root of unity of order p implies that φ(p) ≤ p, where φ is Euler’s function). Show that T is not R+ -rational (use Theorem 2.4) (see Berstel 1971, and also Eilenberg 1974). Show that for any rational series S ∈ KhhAii, where K is a commutative field, the subfield generated by its coefficients is a finitely generated field. Show that if K is a subsemiring of L such that each element in L is a right-linear combination of fixed Pp elements ℓ1 , . . . , ℓp in L, then each Lrational series may be written i=1 ℓi Si for some K-rational series Si (see Lemma II.1.2 and Exercise II.1.3). Show that each Z-rational series is the difference of two N-rational series (use Exercise 3.2). Show that under the hypothesis of Exercise 3.2, if φ is a right K-linear mapping L → K, then fore ach L-rational series S, the series φ(S) = P φ((S, w))w is K-rational. w P Show that for any semiring K, ifS is K n×n -rational, then Si,j = S(w)i,j i,j

is K-rational for fixed i, j in {1, . . . , n} (use Exercise 3.4).

88

Chapter V. Changing the Semiring

Notes to Chapter V Fliess (Fliess 1974a) calls a strong Fatou ring a ring K satisfying Theorem 1.1. Sontag and Rouchaleau (1977) show that for a principal ring K, the ring K[t] is a strong Fatou ring. In the case of one variable, the class of strong Fatou rings is completely characterized by Theorem 1.6. (The formulation is different, but it is equivalent by the results of Section IV.1.) For several variables, a complete characterization of strong Fatou rings is still lacking. A slightly different P proof of Theorem 2.4 (which makes no use of Lemma 2.7) shows that if S = an xn is a rational series with nonnegative coefficients in a subfield K of R, then S is K+ -rational. Thus Theorem 2.10 and Theorem 3.3 give a complete characterization of N-rational series. The proof also shows that any R+ -rational series has star-height at most 2 (over R+ ). The star-height of a rational series S ∈ KhhAii is defined as follows. Consider the sequence R0 ⊂ R1 ⊂ · · · ⊂ Rn ⊂ · · · of sets of series, such that the union of the Rn is the set of all rational series. The set R0 is the set of polynomials, and for S, T ∈ Ri , both S + T and ST are in Ri ; if S ∈ Ri is proper, then S ∗ ∈ Ri+1 . The star-height of a series S is the least integer n with S ∈ Rn . A weak Fatou ring is a ring whose quotient field is a Fatou extensionof it. They are completely characterized in Reutenauer (1980a). These rings have a property which is analogous to the case of one variable, as shown by Cahen and Chabert (1975). Karhum¨ aki (1977) has characterized those polynomials P ∈ Z[x1 , . . . , xn ] such that the rational series over A = {x1 , . . . , xn } X P (|w|x1 , . . . , |w|xn )w w∈A∗

is N-rational.

Chapter VI

Decidability Basically, any operation or property of series can be examined from the aspect of effectivity. In this chapter, we present the positive results, that is those concerning decidable properties, and we limit ourselves to the most striking ones. The first section shows that the most fundamental properties concerning supports (emptiness, finiteness) are decidable. However, most of the other standard questions, such as equality of supports, are undecidable. In the second section, we show first that the size of a finite semigroup of matrices can be bounded (Theorem 2.1). This implies that the finiteness is decidable for a matrix semigroup. As a consequence, one can decide whether the image of a rational series is finite. To complete the chapter, series with polynomial growth are studied. A beautiful characterization (Corollary 2.11) is given, and this property is shown to be decidable.

1

Problems of Supports

We start by showing that several problems concerning the support of rational series are decidable. An instance of these problems is a language L = supp(S), where S is a rational series given by a linear representation (λ, µ, γ) or by a rational expression. The proof of Sch¨ utzenberger’s Theorem I.6.1 shows indeed that a linear representation of a rational series can be effectively computed from one of its rational expressions, and conversely, that a rational expression denoting a recognizable series can be derived from a linear representation. Here we assume that the semiring of coefficients is the field Q (any other “computable” field is also convenient). Proposition 1.1 It is decidable whether the support of a rational series is empty. Proof. This proposition is just a restatement of Corollary II.3.6. Indeed, if L = supp(S), with S a rational series given by a linear representation of dimension n, then the rank of S is less than or equal to n. By the corollary, the series S vanishes if and only if (S, w) = 0 for all words w of length at most n − 1. This condition is easy to test. Finally, it suffices to note the equivalence L = ∅ ⇐⇒ S = 0 . 89

90

Chapter VI. Decidability

 Remark An analogous proof shows that the equality of two rational series is decidable. It suffices to test whether their difference vanishes. Proposition 1.2 It is decidable whether the support of a rational series is finite. Proof. Let L = supp(S). Then L is finite if and only if S is a polynomial. We prove that given a reduced linear representation (λ, µ, γ) of S of dimension n = rank(S), the series S is a polynomial if and only if µw = 0 for all words w of length n. This will prove the proposition since a reduced linear representation is effectively computable as shown in Section II.3. Assume first that µw = 0 for all words of length n. Then the same relation holds for any word w of length ≥ n. Consequently, (S, w) = λµwγ = 0 for all these words, showing that S is a polynomial of degree ≤ n − 1. Conversely, assume that S is a polynomial of degree d, and let w ∈ supp(S) be a word of length d. For any prefix u of w, of length i, the polynomial u−1 S (see Section II.1) has degree d − i. Indeed, w = uv for some word v with |v| = d − i, and (u−1 S, v) = (S, uv) = (S, w) 6= 0, showing that the degree of u−1 S is at least d − i. Moreover, if a word t has length > d − i, then (u−1 S, t) = (S, ut) = 0 since |ut| > d. This shows that u−1 S has degree d − i. It follows that the d+ 1 polynomials of the form u−1 S, where u runs through the prefixes of w, are linearly independent. Consequently dim(S ◦QhAi) ≥ d+1. This dimension is precisely the rank n of the series, by Theorem II.1.5 and Corollary II.1.4. Thus n ≥ d + 1. Consider now a word w of length n. Since |w| > d, one has (S, uwv) = 0 for all words u, v. Consequently the ideal of QhAi generated by w is contained in KerS, and therefore is contained in the syntactic ideal I of S. In particular, w ∈ I. Since I = Kerµ by Corollary II.2.2, it follows that µw = 0.  We now consider an undecidable problem: It is undecidable whether the support of a rational series is the whole free submonoid A∗ . Indeed, setting L = supp(S), one has the equivalence L 6= A∗ ⇐⇒ (S, w) = 0 for some w ∈ A∗ . The claim then follows from the undecidability of the following problem (Tenth problem of Hilbert; theorem of Davis, Putman, Robinson, Matijacevic, Cudnowskii, see Manin 1977, Theorem VI.1.2 et seq.): given a (commutative) polynomial P ∈ Z[x1 , . . . , xn ], does there exist an n-tuple (α1 , . . . , αn ) ∈ Nn such that P (α1 , . . . , αn ) = 0? To such a polynomial P , we indeed associate the series S ∈ Qhhx1 , . . . , xn ii defined for a word w by (S, w) = P (|w|x1 , . . . , |w|xn ) . This series is rational (Example I.5.2). Clearly, there exists a word w with (S, w) = 0 if and only if P vanishes for some n-tuple of nonnegative integers. The undecidability of this problem also implies the undecidability of the following question: Given two supports of rational series, are they equal? See Exercise 1.2 for a proof of these undecidability results using the Post correspondence problem instead of Hilbert’s tenth problem.

2. Growth

2

91

Growth

We first give a result concerning finite monoids of matrices. Recall that for a given word w, we denote by w∗ the submonoid generated by w. Theorem 2.1 (Jacob 1978, Mandel and Simon 1977) Let µ : A∗ → Qn×n be a monoid morphism such that, for all w ∈ A∗ , the monoid µw∗ is finite. Then there exists an effectively computable integer N depending only on |A| and n such that |µ(A∗ )| ≤ N . As we shall see, the function (|A|, n) 7→ N grows extremely rapidly. There exists however one case where there is a reasonable bound (which moreover does not depend on |A|), namely the case described in the lemma below. A set E of matrices in Qn×n is called irreducible if there is no subspace of 1×n Q other than 0 and Q1×n stable for all matrices in E (the matrices act on the right on Q1×n ). Lemma 2.2 Let M ⊂ Qn×n be an irreducible monoid of matrices such that all nonvanishing eigenvalues of matrices in M are roots of unity. Then |M | ≤ 2 (2n + 1)n . Proof. Let m ∈ M . The eigenvalues 6= 0 of m are roots of unity, whence algebraic integers over Z. The same holds for tr(m). Since tr(m) ∈ Q and Z is integrally closed, this implies that tr(m) ∈ Z. The norm of each eigenvalue is 0 or 1. Thus | tr(m)| ≤ n. This shows that tr(m) takes at most 2n + 1 distinct values for m ∈ M . Let m1 , . . . , mk ∈ M be a basis of the subspace N of Qn×n generated by M . Clearly k ≤ n2 . Define an equivalence relation ∼ on M by m ∼ m′ ⇐⇒ tr(mmi ) = tr(m′ mi ) for i = 1, . . . , k . The number of equivalence classes of this relation is at most (2n + 1)k . In order to prove the lemma, it suffices to show that m ∼ m′ implies m = m′ . Let m, m′ ∈ M be such that m ∼ m′ . Set p = m − m′ , and assume p 6= 0. There exists a vector v ∈ Q1×n such that vp 6= 0. It follows that the subspace vpN of Q1×n is not the null space. Since it is stable under M and M is irreducible, one has vpN = Q1×n . Consequently, there exists some q ∈ N such that vpq = v. This shows that pq has the eigenvalue 1. Now, for all integers j ≥ 1, tr((pq)j ) = tr(pq(pq)j−1 ) = 0 because q(pq)j−1 is a linear combination of the matrices m1 , . . . , mk , and by assumption tr(pr) = 0 for r ∈ M . Newton’s formulas show that all eigenvalues of pq vanish. This yields a contradiction.  For the proof of Theorem 2.1, we need another lemma.

92

Chapter VI. Decidability

Lemma 2.3 (Sch¨ utzenberger 1962c) Let µ : A∗ → Qn×n be a morphism into a monoid of matrices which are triangular by blocks  ′  µ ν µ= . 0 µ′′ Assume that µ′ A∗ and µ′′ A∗ are finite, and that µw∗ is finite for any word w. Then X |A|i . |νA∗ | ≤ 0≤i k. The minimal polynomial (see Section IV.1) of the ap xp divides the polynomial xh − xk . Consequently, the roots rational series p∈N

of this series (in the sense defined in Section IV.1) are roots of unity or 0. In view of the unicity of the exponential polynomial (Section IV.2), the λi are roots of unity or 0.

95

2. Growth

Next, if the monoid µA∗ is not irreducible, then µ can be put, by changing the basis, into the form  ′  µ ν µ= 0 µ′′ Arguing by induction, µ is equivalent to a morphism that is triangular by blocks   µ1 ∗ ∗ · · · ∗  µ2 ∗ · · · ∗    µ=  ..  . ∗ 0 0

· · · µn

where each µi A∗ is irreducible. By Lemma 2.2, and according to our computations, all monoids µi A∗ are finite. To complete the proof, it suffices to apply the following lemma iteratively, observing that the product of series with polynomial growth also has polynomial growth.  Lemma 2.9 Let  ′  µ ν µ= 0 µ′′ be a morphism A∗ → K n×n , where K is a commutative semiring. Every series recognized by µ is a linear combination of series recognized by µ′ or by µ′′ and of series of the form S ′ aS ′′ , where S ′ is recognized by µ′ , a ∈ A and S ′′ is recognized by µ′′ .

Proof. A series recognized by µ is a linear combinations of series of the form X (µw)i,j w (2.1) w

with 0 ≤ i, j ≤ n. It suffices to show that when i, j are coordinates corresponding to ν, the series (2.1) is a linear combination of series of the form S ′ aS ′′ . This is a consequence of the formula X νw = µ′ x· νa· µ′′ y . w=xay

 Corollary 2.10 It is decidable whether a rational series S ∈ ZhhAii has polynomial growth. Proof. A reduced linear representation (λ, µ, γ) of S can effectively be computed. Then according to Theorem 2.8, the series S has polynomial growth if and only if the series X tr(µw)w w

has a finite image. This series is rational (Lemma II.1.2) and it is decidable, by Corollary 2.7 whether a rational series has a finite image. 

96

Chapter VI. Decidability

Corollary 2.11 (Sch¨ utzenberger 1962c) The set of Z-rational series of polynomial growth is equal to the Z-subalgebra of ZhhAii generated by the characteristic series of rational languages. Proof. Let S ∈ ZhhAii be a rational series having polynomial growth, and let (λ, µ, γ) be a reduced linear representation of S. We may assume, by Theorem V.1.1, that (λ, µ, γ) has integral coefficients. The second part of the proof of Theorem 2.8 shows that, after a change of the basis of Q1×n , µ has a decomposition of the form   µ1 ∗ ∗ · · · ∗  µ2 ∗ · · · ∗    µ=  ..  . ∗ 0 0 · · · µn

In fact, since Z is a principal ring, the change of basis can be done in Z1×n . Each µi A∗ is finite. Observe now that any series recognized by a morphism µ′ : A∗ → Qp×p with ′ ∗ µ A finite, is a linear combination of characteristic series of rational languages. This is a consequence of Theorem III.2.8. To complete the proof, it suffices to apply Lemma 2.9. Conversely, each characteristic series of a rational language is rational (Proposition III.2.1), and the set of rational series with polynomial growth is indeed a subalgebra of ZhhAii. 

Exercises for Chapter VI 1.1 Show that the following problem is undecidable. Given a rational series S ∈ QhhAii, are there infinitely many words w such that (S, w) = 0? 1.2 Use the undecidability of the Post Correspondence Problem and Exercise III.3.2 to give another proof of the undecidability property of Section 1 (recall that Post’s problem is whether or not a given equality set is empty.) 2.1 Let S ∈ QhhAii be a rational series such that, for every ray R, almost all coefficients (S, w), w ∈ R, vanish. Show that S is a polynomial. 2.2 Let S ∈ NhhAii be an N-rational series having a polynomial growth. Show that S is in the N-subalgebra of NhhAii generated by the characteristic series of rational languages (use a rational expression for S and the fact that if T ∈ NhhAii is not the characteristic series of the basis of a free submonoid of A∗ , then the growth of T ∗ is not polynomial).

Notes to Chapter VI Most of the results of Section 2 hold in arbitrary fields. Theorem 2.1 can be extended, but the bound N then also depends on the field considered. Lemma 2.3 and Corollaries 2.4, 2.5 hold in arbitrary fields, and Lemma 2.2 holds in fields 2 2 of characteristic 0, provided the bound (2n + 1)n is replaced by rn , where r is the size of the set {tr(m) | m ∈ M }. This set is always finite (under the assumptions of the lemma) for a finite monoid M . Corollaries 2.6, 2.7 extend to

97

2. Growth

“computable” fields. Corollary 2.11 is a special case of a result of Sch¨ utzenberger (1962c): for any integer q ≥ 0, the Z-module of rational series S satisfying a condition of the form |(S, w)| ≤ C|w|q

(w ∈ A∗ )

is equal to the Z-module generated by the products of at most q+1 characteristic series of rational languages. He shows also that the degree of growth of a rational series of ZhhAii is always an integer or ∞.

98

Chapter VI. Decidability

Chapter VII

Noncommutative polynomials This chapter deals with algebraic properties of noncommutative polynomials. They are of independent interest, but most of them will be of use in the next chapter. In contrast to commutative polynomials, the algebra of noncommutative polynomials is not Euclidean, and not even factorial. However, there are many interesting results concerning factorization of noncommutative polynomials: this is one of the major topics of the present chapter. The basic tool is Cohn’s weak algorithm (Theorem 1.1) which is the subject of Section 1. This operation constitutes a natural generalization of the classical Euclidean algorithm. Section 2 deals with continuant polynomials which describe the multiplicative relations between noncommutative polynomials (Theorem 2.2). We introduce in Section 3 cancellative modules over the ring of polynomials. We characterize these modules (Theorem 3.1) and obtain, as consequences, results on full matrices, factorization of polynomials, and inertia. The main result of Section 4 is the (easy) extension of Gauss’s lemma to noncommutative polynomials.

1

The Weak Algorithm

Let K be a commutative field and let A be an alphabet. Recall that the degree of a polynomial P in KhAi was defined in Section I.2: we will denote it by deg(P ). We recall the usual facts about the degree, that is deg(0) = −∞ deg(P + Q) ≤ max(deg(P ), deg(Q))

deg(P + Q) = deg(P ), if deg(Q) < deg(P ) deg(P Q) = deg(P ) + deg(Q) .

(1.1) (1.2)

Note that the last equality shows that KhAi is an integral domain, that is P Q = 0 implies

P = 0 or Q = 0 . 99

100

Chapter VII. Noncommutative polynomials

Definition A finite family P1 , . . . , Pn of polynomials in KhAi is (right) dependent if either some Pi = 0 or if there exist polynomials Q1 , . . . , Qn such that X
Pi Qi < max(deg(Pi Qi )) . deg i

i

Definition A polynomial P is (right) dependent family!dependent – on the family P1 , . . . , Pn if either P = 0 or if there exist polynomials Q1 , . . . , Qn such that X
Pi Qi < deg(P ) deg P − i

and if furthermore for any i = 1, . . . , n deg(Pi Qi ) ≤ deg(P ) . Note that if P is dependent on P1 , . . . , Pn then the family P, P1 , . . . , Pn is dependent. The converse is given by the following theorem. Theorem 1.1 (Cohn 1961) Let P1 , . . . , Pn be a dependent family of polynomials with deg(P1 ) ≤ · · · ≤ deg(Pn ) . Then some Pi is dependent on P1 , . . . , Pi−1 . Let P be a polynomial and let u be a word in A∗ . We define the polynomial P u as X P u−1 = (P, wu)w . −1

w∈A∗

The operator P 7→ P u−1 is symmetric to the operator P 7→ u−1 P which was introduced in Section I.5. It is easy to verify that this operator is linear, and that the following relations hold: deg(P u−1 ) ≤ deg(P ) − |u| −1

P (uv)

= (P v

−1

)u

−1

(1.3) (1.4)

Moreover, for any letter a, (P Q)a−1 = P (Qa−1 ) + (Q, 1)P a−1

(1.5)

where (Q, 1) denotes as usual the constant term of Q. The last equality is simply the symmetric equivalent of Lemma I.6.2. Lemma 1.2 If P, Q are polynomials and w is a word, then there exists a polynomial P ′ such that (P Q)w−1 = P (Qw−1 ) + P ′ with either P = P ′ = 0 or deg(P ′ ) < deg(P ).

101

1. The Weak Algorithm

Proof. We may assume P 6= 0. If w is the empty word, then (P Q)w−1 = P Q and Qw−1 = Q, so that (P Q)w−1 = P (Qw−1 ) and the proof is complete. Let w = au with a a letter. Then by induction one has (P Q)u−1 = P (Qu−1 ) + P ′ deg(P ′ ) < deg(P ) Now, by Eq. (1.4), one has (P Q)w−1 = ((P Q)u−1 )a−1 = (P (Qu−1 ))a−1 + P ′ a−1 . Thus, by Eqs.(1.5) and (1.4), we have (P Q)w−1 = P ((Qu−1 )a−1 ) + (Qu−1 , 1)P a−1 + P ′ a−1 = P (Qw−1 ) + P ′′ with P ′′ = (Qu−1 , 1)P a−1 + P ′ a−1 . Next, by Eq. (1.3), deg(P a−1 ) < deg(P ) and deg(P ′ a−1 ) ≤ deg(P ′ )−|a| < deg(P ). Hence deg(P ′′ ) < deg(P ), as desired.  ProofP of Theorem 1.1. We may suppose that no Pi is equal to 0. Hence deg( Pi Qi ) < maxi (deg(Pi Qi )). Let r P = maxi (deg(Pi Qi )) and let I = {i | Pi Qi has degree deg(R) < r. Let deg(Pi Qi ) = r}. The polynomial R = i∈I

k = sup(I); then i ∈ I =⇒ deg(Pi ) ≤ deg(Pk ). Let w be a word such that |w| = deg(Qk ) and 0 6= (Qk , w) = α−1 ∈ K: such a word exists because Qk 6= 0 (otherwise deg(R) < r = deg(Pk Qk ) = −∞). By Lemma 1.2, we have X X Rw−1 = Pi (Qi w−1 ) + Pi′ i∈I

i∈I

for some polynomials Pi′ with deg(Pi′ ) < deg(Pi ). Since Qk w−1 = α−1 , X X Pi (Qi w−1 ) = αRw−1 − α Pi′ . Pk + α i∈I\k

(1.6)

i∈I

Now, by Eq. (1.3) deg(Rw−1 ) ≤ deg(R) − |w| < r − |w| = deg(Pk Qk ) − deg(Qk ) = deg(Pk ) . Furthermore, deg(Pi′ ) < deg(Pi ) ≤ deg(Pk ). Consequently, by Eq. (1.1), the degree of the right-hand side of Eq. (1.6) is < deg(Pk ). Moreover, deg(Pi (Qi w−1 )) = deg(Pi ) + deg(Qi w−1 ) ≤ deg(Pi ) + deg(Qi ) − deg(Qk ) by Eq. (1.3). So we have deg(Pi (Qi w−1 )) ≤ r − deg(Qk ) = deg(Pk ). This shows that Pk is dependent on Pi , i ∈ I \k; hence Pk also is dependent on P1 , . . . , Pk−1 .  For two polynomials X, Y in KhAi, the (left) Euclidean division of X and Y (that is the problem of finding polynomials Q and R such that X = Y Q + R and deg(R) < deg(Y )) is not always possible. However, the next result gives a necessary and sufficient condition for this.

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Corollary 1.3 Let X, Y, P, Q1 , Q2 , R1 be polynomials such that XP + Q1 = Y Q2 + R1 with P 6= 0, deg(Q1 ) ≤ deg(P ), deg(R1 ) < deg(Y ) . Then there exists polynomials Q and R such that X = Y Q + R with deg(R) < deg(Y ) (that is, Euclidean division of X by Y is possible). Proof. Note that Y 6= 0 (otherwise deg(R1 ) < −∞). If Y ∈ K, the corollary is immediate (take Q = Y −1 X and R = 0). Otherwise, we prove it by induction on deg(X). If deg(X) < deg(Y ), the proof is immediate (take Q = 0 and R = X). Suppose that deg(X) ≥ deg(Y ). Then deg(Q1 ) ≤ deg(P ) < deg(XP ) because 1 ≤ deg(Y ) ≤ deg(X) and deg(R1 ) < deg(Y ) ≤ deg(X) ≤ deg(XP ) because 0 ≤ deg(P ). Thus, deg(Q1 ) and deg(R1 ) are both < max(deg(XP ), deg(Y Q2 )) and by Eq. (1.1), deg(R1 − Q1 ) < max(deg(XP ), deg(Y Q2 )). In view of Theorem 1.1, X is dependent on Y , that is there exist two polynomials Q3 and X1 such that X = Y Q3 + X1 with deg(X1 ) < deg(X). Put this expression for X into the initial equality. This gives X1 P + Q1 = Y (Q2 − Q3 P ) + R1 . Since deg(X1 ) < deg(X), we have by induction X1 = Y Q4 + R with deg(R) < deg(Y ). Thus X = Y Q3 + Y Q4 + R, which proves the corollary.  The next result is a particular case of the previous one. Corollary 1.4 If X, Y, X ′ , Y ′ are nonzero polynomials such that XY ′ = Y X ′ , then there exist polynomials Q, R such that X = Y Q + R and deg(R) < deg(Y ). 

2

Continuant Polynomials

Definition Let a1 , . . . , an be a finite sequence of polynomials. We define the sequences p0 , . . . , pn of continuant polynomials (with respect to a1 , . . . , an ) in the following way: p0 = 1, p1 = a1 , and for 2 ≤ i ≤ n, pi = pi−1 ai + pi−2 .

2. Continuant Polynomials

103

Example 2.1 The first continuant polynomials are p 2 = a1 a2 + 1 p 3 = a1 a2 a3 + a1 + a3 p 4 = a1 a2 a3 a4 + a1 a2 + a1 a4 + a3 a4 + 1 Notation We shall write p(a1 , . . . , ai ) for pi . It is easy to see that the continuant polynomials may be obtained by the “leap-frog construction”: consider the “word” a1 · · · an and all words obtained by repetitively suppressing some factors of the form ai ai+1 in it. Then p(a1 , . . . , an ) is the sum of all these “words”. Now, we have by definition p(a1 , . . . , an ) = p(a1 , . . . , an−1 )an + p(a1 , . . . , an−2 ) .

(2.1)

The combinatorial construction sketched above shows that symmetrically p(a1 , . . . , an ) = a1 p(a2 , . . . , an ) + p(a3 , . . . , an ) .

(2.2)

An equivalent but useful relation is p(an , . . . , a1 ) = an p(an−1 , . . . , a1 ) + p(an−2 , . . . , a1 ) .

(2.3)

Proposition 2.1 (Wedderburn 1932) The continuant polynomials satisfy the relation p(a1 , . . . , an )p(an−1 , . . . , a1 ) = p(a1 , . . . , an−1 )p(an , . . . , a1 ) .

(2.4)

Proof. This is surely true for n = 1. Suppose n ≥ 2. Then by Eq. (2.1), p(a1 , . . . , an )p(an−1 , . . . , a1 ) = p(a1 , . . . , an−1 ) an p(an−1 , . . . , a1 ) + p(a1 , . . . , an−2 )p(an−1 , . . . , a1 ) which is equal by induction to p(a1 , . . . , an−1 ) an p(an−1 , . . . , a1 ) + p(a1 , . . . , an−1 )p(an−2 , . . . , a1 ) . This is equal, by Eq. (2.3), to p(a1 , . . . , an−1 )p(an , . . . , a1 ) as desired.  Theorem 2.2 (Cohn 1969) Let X, Y, X ′ , Y ′ be nonzero polynomials such that XY ′ = Y X ′ . Then there exists polynomials U, V, a1 , . . . , an with n ≥ 1 such that X = U p(a1 , . . . , an ), Y ′ = p(an−1 , . . . , a1 )V Y = U p(a1 , . . . , an−1 ), X ′ = p(an , . . . , a1 )V . Moreover, one has deg(a1 ), . . . , deg(an−1 ) ≥ 1, and if deg(X) > deg(Y ), then deg(an ) ≥ 1.

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Proof. (i) Suppose first that X is a right multiple of Y , that is X = Y Q. Then the theorem is obvious for U = Y , V = Y ′ , n = 1, a1 = Q; then indeed X = Y Q = U p(a1 ), Y ′ = 1 · V, Y = U · 1 and Y X ′ = XY ′ = Y QY ′ , whence X ′ = QY ′ = p(a1 )V . Furthermore, if deg(X) > deg(Y ), then deg(Q) ≥ 1. (ii) Next, we prove the theorem in the case where deg(X) > deg(Y ), by induction on deg(Y ). If deg(Y ) = 0, then X is a right multiple of Y and we may apply (i). Suppose deg(Y ) ≥ 1. By Corollary 1.4, X = Y Q + R for some polynomials Q and R such that deg(R) < deg(Y ). If R = 0, apply (i). Otherwise, we have Y X ′ = XY ′ = Y QY ′ + RY ′ , hence Y (X ′ − QY ′ ) = RY ′ ; note that Y, R, Y ′ 6= 0, hence X ′ − QY ′ 6= 0. Furthermore, deg(R) < deg(Y ), and we may apply the induction hypothesis: there exist polynomials U, V, a1 , . . . , an such that Y = U p(a1 , . . . , an ), X ′ − QY ′ = p(an−1 , . . . , a1 )V

R = U p(a1 , . . . , an−1 ), Y ′ = p(an , . . . , a1 )V deg(a1 ), . . . , deg(an ) ≥ 1 .

(2.5)

Hence
X = Y Q + R = U p(a1 , . . . , an )Q + p(a1 , . . . , an−1 ) = U p(a1 , . . . , an , Q)

by Eq. (2.1). Similarly, X ′ = p(Q, an , . . . , a1 )V . Thus X, Y, X ′ , Y ′ admit the announced expression. Furthermore, deg(Q) ≥ 1; indeed, by Eq. (1.2), deg(X) = deg(Y Q) = deg(Y ) + deg(Q), and hence deg(Q) = deg(X) − deg(Y ) ≥ 1. This prove the theorem in the case where deg(X) > deg(Y ). (iii) In the general case, one has again X = Y Q + R with deg(R) < deg(Y ) (Corollary 1.4). If R = 0, the proof is completed by (i). Otherwise, as above, Y (X ′ − QY ′ ) = RY ′ with deg(Y ) > deg(R). Hence we may apply (ii): there exist U, V, a1 , . . . , an such that Eq. (2.5) holds. Then we obtain, as in (ii): X = U p(a1 , . . . , an , Q), Y ′ = p(an , . . . , a1 )V Y = U p(a1 , . . . , an ), X ′ = p(Q, an , . . . , a1 )V . This proves the theorem.  Proposition 2.3 Let a1 , . . . , an be polynomials such that a1 , . . . , an−1 have positive degree, and let Y be a polynomial of degree 1 such that p(an−1 , . . . , a1 ) and p(an . . . , a1 ) are both congruent to a scalar modulo the right ideal Y KhAi. Then for i = 1, . . . , n p(ai , . . . , a1 ) ≡ p(a1 , . . . , ai )

mod Y KhAi .

We prove first a lemma. Lemma 2.4 Let a1 , . . . , an be polynomials such that a1 , . . . , an−1 have positive degree. Then the degrees of 1, p(a1 ), . . . , p(an−1 , . . . , a1 ) are strictly increasing.

105

2. Continuant Polynomials Proof. Obviously deg(1) < deg(a1 ). Suppose deg(p(ai−2 , . . . , a1 )) < deg(p(ai−1 , . . . , a1 )) for 2 ≤ i ≤ n − 1. From the relation p(ai , . . . , a1 ) = ai p(ai−1 , . . . , a1 ) + p(ai−2 , . . . , a1 ) ,

it follows that the degree of p(ai , . . . , a1 ) is equal to deg(ai p(ai−1 , . . . , a1 )), and deg(ai p(ai−1 . . . , a1 )) = deg(ai ) + deg(p(ai−1 , . . . , a1 )) > deg(p(ai−1 , . . . , a1 )) because deg(ai ) ≥ 1. This proves the lemma.  Proof of Proposition 2.3 (Induction on n). When n = 1, the result is evident. Suppose n ≥ 2. Note that if the condition on the degrees is fulfilled for a1 , . . . , an , then a fortiori also a1 , . . . , an−2 have positive degree. By assumption, p(an , . . . , a1 ) is congruent to some scalar α and p(an−1 , . . . , a1 ) is congruent to some scalar β mod. Y KhAi. Suppose p(an−1 , . . . , a1 ) = 0. Then by Eq. (2.3), we have p(an−2 , . . . , a1 ) ≡ α = α − βγ for any γ, because β = 0 in this case. Suppose p(an−1 , . . . , a1 ) 6= 0. Then by Eq. (2.3), an p(an−1 , . . . , a1 ) + p(an−2 , . . . , a1 ) = Y Q + α for some polynomial Q. As deg(p(an−2 , . . . , a1 )) < deg(p(an−1 , . . . , a1 )) by Lemma 2.4, we obtain by Corollary 1.3 that an ≡ γ mod Y KhAi for some scalar γ. Using Eq. (2.3) again, and the fact that P ≡ γ, Q ≡ β =⇒ P Q ≡ γβ, we obtain p(an−2 , . . . , a1 ) ≡ α − γβ. In both cases, the induction hypothesis gives p(a1 , . . . , an−2 ) ≡ α − γβ and p(a1 , . . . , an−1 ) ≡ β. Hence, by Eq. (2.1), p(a1 , . . . , an ) ∈ (β + Y KhAi)(γ + Y KhAi) + α − βγ + Y KhAi, and consequently p(a1 , . . . , an ) ≡ βγ + α − γβ ≡ p(an . . . , a1 ), as desired.  Lemma 2.5 Let a1 , . . . , an be polynomials. Then p(a1 , . . . , an ) = 0 ⇐⇒ p(an , . . . , a1 ) = 0 . Proof (Induction on n). The lemma is evidently true for n = 0, 1. Suppose n ≥ 2. It is enough to show that p(a1 , . . . , an ) = 0 implies p(an , . . . , a1 ) = 0. Now, by Eq. (2.4), p(a1 , . . . , an )p(an−1 , . . . , a1 ) = p(a1 , . . . , an−1 )p(an , . . . , a1 ) . Suppose p(a1 , . . . , an ) = 0. If p(a1 , . . . , an−1 ) 6= 0, then p(an , . . . , a1 ) = 0 because KhAi is an integral domain. If p(a1 , . . . , an−1 ) = 0, then p(an−1 , . . . , a1 ) = 0 by induction. Hence, by Eqs. (2.1) and (2.3) p(a1 , . . . , an ) = p(a1 , . . . , an−2 ) and p(an , . . . , a1 ) = p(an−2 , . . . , a1 ). By induction, p(a1 , . . . , an−2 ) and p(an−2 , . . . , a1 ) simultaneously vanish, which proves the lemma. 

106

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Chapter VII. Noncommutative polynomials

Inertia

Recall that KhAip×q denotes the set of p by q matrices over KhAi. In particular, KhAin×1 is the set of column vectors of order n over KhAi. This set has a natural structure of right KhAi-module. If V is in KhAin×1 , we denote by (V, 1) its constant term, that is, setting   P1  ..  V = .  Pn

one has



 (P1 , 1)   (V, 1) =  ...  ∈ KhAin×1 . (Pn , 1)

Furthermore, if w is a word in A∗ , we denote by V w−1 the vector   P1 w−1   V w−1 =  ...  . Pn w−1

We have the following relation X V = (V, 1) + (V a−1 )a .

(3.1)

a∈A

Definition A (right) submodule E of KhAin×1 is cancellative if, whenever V ∈ E and (V, 1) = 0, then V a−1 ∈ E for any letter a ∈ A. This property of vectors of polynomials is closely related to (but weaker than) the property of stability introduced in Section I.5. The next result characterizes cancellative submodules and will be the key to all the results of this section. Theorem 3.1 A submodule E of KhAin×1 is cancellative if and only if it may be generated, as a right KhAi-module, by p vectors V1 , . . . , Vp such that the matrix ((V1 , 1), . . . , (Vp , 1)) ∈ K n×p is of rank p. In this case, p ≤ n and V1 , . . . , Vp are linearly KhAi-independent. Proof. 1. We begin with the easy part: suppose that E is generated by V1 , . . . , Vp as indicated. Let V ∈ E with (V, 1) = 0. Then V =

X

1≤i≤p

Vi Pi

(Pi ∈ KhAi) .

Taking constant terms, we obtain X 0 = (V, 1) = (Vi , 1)(Pi , 1) .

107

3. Inertia

Because of the rank condition, we have (Pi , 1) = 0 for any i. Hence Pi = P (Pi a−1 )a, which shows that a∈A

V =

X

Vi (Pi a−1 )a .

i, a

By Eq. (3.1) we obtain X Vi (Pi a−1 ) . V a−1 = i

hence V a−1 ∈ E, as desired. 2. Let E be If V ∈ KhAin×1 , V may be P a cancellative submodule of KhAi.n×1 are almost all zero. Let written V = w∈A∗ (V, w)w where (V, w) ∈ KhAi deg(V ) be the maximal length of a word w such that (V, w) 6= 0. Claim. There are vectors V1 , . . . , Vp in E such that

(i) deg(V1 ) ≤ deg(V2 ) ≤ · · · ≤ deg(Vp ) . (ii) The vectors (Vi , 1) form a K-basis of the K-space (E, 1) = {(V, 1) | V ∈ E}. (iii) If V ∈ E and deg(V ) < deg(Vi ) then (V, 1) is a K-linear combination of (V1 , 1), . . . , (Vi−1 , 1). Suppose the claim is true. Then the matrix ((V1 , 1), . . . , (VP p , 1)) has rank p. We show by induction on deg(V ) that each V ∈ E is in E ′ = 1≤i≤p Vi KhAi. If deg(V ) = −∞, that is V = 0, it is obvious. Let deg(V ) ≥ 0 and let i be the smallest integer such that deg(V ) < deg(Vi ) (with i = p + 1 if such an integer does not exist). Then deg(V ) ≥ deg(V1 ), . . . , deg(Vi−1 ). Moreover, if i ≤ p then by (iii) (V, 1) is a linear combination of (V1 , 1), . . . , (Vi−1 , 1), and if i = p + 1 then byP(ii), (V, 1) is also a linear combination of (V1 , 1), . . . , (Vi−1 , 1). Let V ′ = V − 1≤j≤i−1 αj Vj (αj ∈ K) be such that (V ′ , 1) = 0. By the cancellative property of E, V ′ a−1 is in E for any letter a. Now, deg(V ′ ) ≤ max(deg(V ), deg(α1 V1 ), . . . , deg(αi−1 Vi−1 )) = deg(V ) hence deg(V ′ a−1 )P< deg(V ). Hence by induction, V ′ a−1 ∈ E ′ . PNow, by ′ −1 )a, and V ′ is in E ′ . Thus V = V ′ + αj Vj is Eq. (3.1), V ′ = a (V a j

in E ′ as well. 3. Proof of the claim. For d = −1, 0, 1, 2, . . ., let F (d) be the subspace of K n×1 defined by F (d) = {(V, 1) | V ∈ E, deg(V ) ≤ d} . Then 0 = F (−1) ⊂ F (0) ⊂ F (1) ⊂ · · · ⊂ F (d) ⊂ · · · Let 0 ≤ d1 < · · · < dq be such that for any i, F (di − 1) ( F (di ) and such that each F (d) is equal to some F (di ); in other words, one has 0 = F (−1) = · · · = F (d1 − 1) ( F (d1 ) = · · · = F (d2 − 1) ( F (d2 ) ( · · · ( F (dq ) = F (dq + 1) = · · ·

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Chapter VII. Noncommutative polynomials

In particular, F (dq ) = (E, 1). Now, let B1 be a basis of F (d1 ), B2 be a basis of F (d2 ) mod F (d1 ), . . . , and let Bq be a basis of F (dq ) mod F (dq−1 ). By the definition of the F ’s we may find for each i in {1, . . . , q} vectors Wi,1 , . . . , Wi,ki in E of degree ≤ di such that {(Wi,1 , 1), . . . , (Wi,ki , 1)} = Bi ; in fact, the degree of each Wi,j is exactly di , otherwise (Wi,j , 1) ∈ F (di − 1) = F (di−1 ), which contradicts the fact that Bi is a basis mod F (di−1 ). Define V1 , . . . , Vp by (V1 , . . . , Vp ) = (W1,1 , . . . , W1,k1 , W2,1 , . . . , W2,k2 , . . . , Wq,kq ) . Then the condition (i) of the claim is clearly satisfied. Moreover, as F (dq ) = (E, 1), condition (ii) is also satisfied. Let V ∈ E with deg(V ) < deg(Vk ). Then Vk = Wi,j for some i, j, hence deg(V ) < di = deg(Wi,j ), which implies that (V, 1) ∈ F (di − 1) = F (di−1 ) and (V, 1) is a linear combination of W1,1 , . . . , Wi−1,ki−1 , hence of V1 , . . . , Vk−1 . This proves the claim. P 4. We show the last assertion of the theorem. Clearly, p ≤ n. Suppose Vi Pi = 0 where Pi ∈ KhAiPare not all zero; choose such a relation with sup(deg(Pi )) minimum. Then (Vi , 1)(Pi , 1) = 0 which shows as in (1) that (Pi , 1) = 0 for each i. Now Pj is 6= 0, hence Pj a−1 6= 0 for some letter a. P some −1 By Eq. (3.1) we obtain Vi (Pi a ) = 0, which is a new relation contradicting the above minimality. Thus the V ’s are KhAi-independent.  Definition An n by n matrix M over KhAi is full if, whenever M = M1 M2 for some matrices M1 ∈ KhAin×p and M2 ∈ KhAip×n , then p ≥ n. Remark Taking in the above definition a field instead of KhAi, one obtains exactly the definition of an invertible matrix over this field. Corollary 3.2 (Cohn 1961) Let M be an n by n matrix over KhAi. If S1 , . . . , Sn in KhhAii are formal series, not all zero, such that (S1 , . . . , Sn )M = (0, . . . , 0), then M is not full. Proof. Let E be the set of vectors V ∈ KhAin×1 such that (S1 , . . . , Sn )V = 0. Then E is a right submodule of KhAin×1 . Let V = t (P P1 , . . . , Pn ) ∈ E be such that (V, 1) = 0. Then (Pi , 1) = 0Pfor any i. Moreover i Si Pi = 0, so that if a is a letter, by Eq. (3.1), one has i Si (Pi a−1 ) = 0. This means that V a−1 ∈ E; thus E is cancellative. By Theorem 3.1, the right KhAi-module E admits a basis consisting of p vectors V1 , . . . , Vp such that rank((V1 , 1), . . . , (Vp , 1)) = p and p ≤ n. Now suppose that p = n. Then the matrix N = ((V1 , 1), . . . , (Vn , 1)) ∈ K n×n is invertible. But N is the constant matrix of H = (V1 , . . . , Vn ) ∈ KhAin×n , that is N = (H, 1); this implies that H is invertible in KhhAiin×n . Now we have (S1 , . . . , Sn )H = 0 (because (S1 , . . . , Sn )Vi = 0 for all i), hence (S1 , . . . , Sn ) = 0 (multiply by H −1 ), a contradiction. So p < n. Let M = (C1 , . . . , Cn ), where Ck is the k-th column of M . Then, p P Vj Pj,k for some polynomials by hypothesis, Ck belongs to E, hence Ck = j=1

Pj,k . Thus

M = (V1 , . . . , Vp )(Pj,k )1≤j≤p, 1≤k≤n and M is not full. 

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3. Inertia

Corollary 3.3 (Cohn 1982) Let P1 , P2 , P3 , P4 be polynomials such that P2 is invertible as a formal series, that is (P2 , 1) 6= 0, and such that P1 P2−1 P3 = P4 holds in KhhAii. Then there exist polynomials Q1 , Q2 , Q3 , Q4 such that P1 = Q1 Q2 , P2 = Q3 Q2 , P3 = Q3 Q4 , P4 = Q1 Q4 . Proof. Consider the 2 by 2 matrix over KhAi:   P1 P4 M= P2 P3 By assumption, we have (1, −P1 P2−1 )M = 0 . Hence M is not full by Corollary 3.2, and M may be written as   Q1 (Q2 , Q4 ) M= Q3 for some polynomials Qi . This proves the corollary.  The next result is the Inertia Theorem. It will not be used in Chapter VIII. Let S1 , . . . , Sn , T1 , . . . , Tn be formal series. We say that X Sj T j j

is trivially a polynomial if, for each j, either Sj = 0, or Tj = 0, or both Sj and Tj are polynomials. Note that one has   T1 X   Sj Tj = (S1 , . . . , Sn )  ...  . j Tn

Corollary 3.4 (Inertia Theorem, Bergmann 1967, Cohn 1961) Let (Si,h )i∈I, 1≤h≤n and (Th,j )1≤h≤n, P j∈J be two families of formal series such that for each i ∈ I and j ∈ J, h Si,h Th,j is a polynomial. Then there exists an invertible matrix M over KhhAii such that for any i and j    T1,j    −1  .  (Si,1 , . . . , Si,n )M M  ..  Tn,j

is trivially a polynomial.

Proof. 1. We prove the theorem first in the case where each Th,j is a polynomial. Let E = {V ∈ KhAin×1 | ∀i ∈ I, (Si,1 , . . . , Si,n )V ∈ KhAi}. Then E is a cancellative right submodule of KhAin×1 as may be easily verified (cf. the proof of Corollary 3.2). By Theorem 3.1 there exist p vectors V1 , . . . , Vp in E which form a basis of E (as a right KhAi-module) and such that the constant matrix of (V1 , . . . , Vp ) is of rank p ≤ n. By performing a permutation of coordinates, we may assume that   X (V1 , . . . , Vp ) = , Y

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Chapter VII. Noncommutative polynomials

where (X, 1) ∈ K p×p is invertible. Let   X 0 , M= Y In−p where In−p is the identity matrix of order n − p. Then (M, 1) ∈ K n×n is invertible, hence M is invertible in KhhAiin×n . Note that the first p columns of M (that is the Vi ’s) are in E: this implies, by definition of E, that for any i ∈ I the first p components of (Si,1 , . . . , Si,n )M are polynomials. Moreover, let 1 ≤ h ≤ p: then M −1 Vh is equal to the hth column of M −1 M , that is to the hth canonical vector Eh ∈ K n×1 . PNow let j ∈ J. Then by assumption V = t (T1,j , . . . , Tn,j ) is in E. Hence V = 1≤h≤p Vh Ph for P some polynomials Ph . Thus M −1 V = h M −1 Vh Ph is equal, by the previous P remark, to h Eh Ph = t (P1 , . . . , Pp , 0, . . . , 0). This shows that the product    T1,j    −1  .  (Si,1 , . . . , Si,n )M M  ..  Tn,j

is trivially a polynomial. 2. We come to the general case. Let

H = {h ∈ {1, . . . , n} | ∀j ∈ J, Th,j ∈ KhAi} . If H = {1, . . . , n}, then we are in case 1. Suppose |H| < n: we may suppose that H = {1, . . . , p} with 0 ≤ p < n (including the case H = ∅). Suppose that ∀i ∈ I, ∀h ∈ / H, Si,h = 0. Then n X

Si,h Th,j =

h=1

p X

Si,h Th,j

h=1

is a polynomial, so we are also in case 1 (with p instead of n). Otherwise, there is some i0 ∈ I such that for some h0 ∈ / H, Si0 ,h0 6= 0. Choose h0 ∈ / H such that ω(Si0 ,h0 ) ≤ ω(Si0 ,h ) for any h ∈ / H (for the definition of ω, see Section I.3). Choose polynomials R1 , . . . , Rp such that for 1 ≤ h ≤ p, ω(Si0 ,h + Rh ) ≥ ω(Si0 ,h0 ). Define Sh′ by Sh′ = Si0 ,h + Rh if 1 ≤ h ≤ p and Sh′ = Si0 ,h if p < h ≤ n. Then ω(Sh′ 0 ) ≤ ω(Sh′ ), Sh′ 0 = Si0 ,h0 6= 0 and X

Sh′ Th,j =

1≤h≤n

X

(Si0 ,h + Rh )Th,j +

h≤p

=

X

1≤h≤n

X

Si0 ,h Th,j

h>p

Si0 ,h Th,j +

X

Rh Th,j

h≤p

is a polynomial, by definition of H = {1, . . . , p}. Let w be a word of minimal length in the support of Sh′ 0 ; then w−1 Sh′ 0 is an invertible formal series, and for any h, since ω(Sh′ ) ≥ |w|, one has w−1 (Sh′ Th,j ) = (w−1 Sh′ )Th,j . Hence P −1 ′ Sh )Th,j is a polynomial. Define the matrix N ∈ KhhAiin×n which coinh (w cides with the n × n identity matrix except in the h0 th row, where it is equal to (w−1 S1′ , . . . , w−1 Sn′ ); in particular the entry of the coordinate (h0 , h0 ) of N is the invertible series w−1 Sh′ 0 , so N is invertible in KhhAiin×n . Let M = N −1 . Then

4. Gauss’s Lemma

111

for any j, M −1 t (T1,j , . . . , Tn,j ) = N t (T1,j , . . . , Tn,j ) is to t (T1,j , . . . , Tn,j ) Pequal −1 ′ except in the h0 th component, where it is equal to (w Sh )Th,j : hence the first p and the h0 th components of M −1 t (T1,j , . . . , Tn,j ) are polynomials and we may conclude the proof by induction on n − p because we have increased |H|. 

4

Gauss’s Lemma

We consider in this section polynomials with integer or rational coefficients. Everything would work, however, with any factorial ring instead of Z. Definition A polynomial P ∈ QhAi is primitive if P 6= 0, P ∈ ZhAi and if its coefficients have no nontrivial common divisors in Z. Definition The content of a nonzero polynomial P ∈ QhAi is the unique positive rational number c(P ) such that P/c(P ) is primitive. Notation P/c(P ) will be denoted by P . Example 4.1 c(4/3+6a−2ab) = 2/3 because 3/2(4/3+6a−2ab) = 2+9a−3ab is primitive. Note that for P 6= 0 P primitive ⇐⇒ c(P ) = 1 P ∈ ZhAi ⇐⇒ c(P ) ∈ N .

(4.1) (4.2)

Theorem 4.1 (Gauss’s Lemma) (i) If P, Q are primitive, then so is P Q. (ii) If P, Q are nonzero polynomials, then c(P Q) = c(P )c(Q) and P Q = P Q. Proof (i) Suppose P Q is not primitive. Then there is some prime number n which divides each coefficient of P Q. This means that the canonical image φ(P Q) of P Q in (Z/nZ)hAi vanishes. But Z/nZ is a field, so (Z/nZ)hAi is an integral domain (Section I.1); moreover 0 = φ(P Q) = φ(P )φ(Q), so φ(P ) = 0 or φ(Q) = 0. This means that n divides all coefficients of P or of Q, and contradicts the fact that P and Q are primitive. (ii) By (i), P Q/c(P )c(Q) = (P/c(P ))(Q/c(Q)) is primitive. So, by definition of the content of P Q, c(P Q) = c(P )c(Q). Now, P Q = P Q/c(P Q) so that P Q = P Q/c(P )c(Q) = P Q.  Corollary 4.2 Let a1 , . . . , an be polynomials. Then the continuant polynomials p(a1 , . . . , an ) and p(an , . . . , a1 ) are both zero or have the same content. Proof (Induction on n). The result is obvious for n = 0, 1. Let n ≥ 2. By Lemma 2.5, we may suppose that both polynomials are 6= 0. Now we have, by Proposition 2.1 p(a1 , . . . , an )p(an−1 , . . . , a1 ) = p(a1 , . . . , an−1 )p(an , . . . , a1 ) .

112

Chapter VII. Noncommutative polynomials

By induction, either p(a1 , . . . , an−1 ) = p(an−1 , . . . , a1 ) = 0, in which case p(a1 , . . . , an ) = p(a1 , . . . , an−2 ) by Eq. (2.1) and p(an , . . . , a1 ) = p(an−2 , . . . , a1 ) and we conclude by induction; or c(p(an−1 , . . . , a1 )) = c(p(a1 , . . . , an−1 )), which implies by Eq. (2.4) and Theorem 4.1 that c(p(a1 , . . . , an )) = c(p(an , . . . , a1 )).  Corollary 4.3 Let P1 , P2 , P3 , P4 be nonzero polynomials in ZhAi such that P2 is invertible in QhhAii and such that P1 P2−1 P3 = P4 . Then there exist polynomials R1 , R2 , R3 , R4 ∈ ZhAi such that P1 = R1 R2 , P2 = R3 R2 , P3 = R3 R4 , P4 = R1 R4 . Proof. By Corollary 3.3 we have P1 = Q1 Q2 , P2 = Q3 Q2 , P3 = Q3 Q4 , P4 = Q1 Q4 for some polynomials Q1 , Q2 , Q3 , Q4 ∈ QhAi. Let ci = c(Qi ), i = 1, 2, 3, 4. By Theorem 4.1 we have c(P1 ) = c1 c2 , c(P2 ) = c3 c2 , c(P3 ) = c3 c4 , c(P4 ) = c1 c4 . Thus c(P4 ) = c(P1 )c(P3 )/c(P2 ). As by hypothesis and Eq. (4.2) c(Pi ) ∈ N, there exist positive integers d1 , d2 , d3 , d4 such that c(P1 ) = d1 d2 , c(P2 ) = d3 d2 , c(P3 ) = d3 d4 , c(P4 ) = d1 d4 . Moreover, by Theorem 4.1, P 1 = Q1 Q2 , P 2 = Q3 Q2 , P 3 = Q3 Q4 , P 4 = Q1 Q4 . Put Ri = di Qi , i = 1, 2, 3, 4. Then Ri ∈ ZhAi. Moreover P1 = c(P1 )P 1 = d1 d2 Q1 Q2 = R1 R2 . Similarly P2 = R3 R2 , P3 = R3 R4 and P4 = R1 R4 .  Proposition 4.4 Let Y be a primitive polynomial of degree 1 which vanishes for some integer values of the variables. Let P, Q ∈ ZhAi and let α ∈ Z, α 6= 0 be such that P Q ≡ α mod Y ZhAi. Then P ≡ β, Q ≡ γ mod Y ZhAi for some β, γ ∈ Z such that α = βγ. Proof. We have P Q = Y Q2 + α for some polynomial Q2 . As α 6= 0, we have Q 6= 0 and we may apply Corollary 1.3. This shows that P = β + Y T for some β ∈ Q and T ∈ QhAi. Hence Y Q2 + α = βQ + Y T Q. Since α 6= 0 and deg(Y ) > 0, we obtain β 6= 0: indeed, otherwise P = Y T and Y T Q = Y Q2 + α, implying that Y divides α. This shows that Q = γ + Y S for some γ ∈ Q such that α = βγ. Now the assumption on Y and the fact that P, Q have integer coefficients imply that β, γ ∈ Z. Since Y T = P − β ∈ ZhAi, we obtain that c(Y )c(T ) ∈ N by Eq. (4.2) and Theorem 4.1 (ii). But Y is primitive, so c(Y ) = 1, which shows that c(T ) ∈ N and T ∈ ZhAi by (4.2). Similarly, S ∈ ZhAi. 

113

4. Gauss’s Lemma

Exercises for Chapter VII n P

Pi Qi = 0 is P called trivial if for each i, either Pi = 0 or Qi = 0. Note that Pi Qi may be written   Q1  ..  (P1 , . . . , Pn )  .  .

1.1 Let P1 , . . . , Pn , Q1 , . . . , Qn be polynomials. A relation

i=1

Qn

Show that if

n P

Pi Qi = 0, then there exists an invertible n by n matrix

i=1

M with coefficients in KhAi such that the relation    Q1    −1  .  (P1 , . . . , Pn )M M  ..  = 0 Qn

is trivial (cf. Cohn 1961). 1.2 a) Let X, Y X ′ , Y ′ be nonzero formal series such that XY ′ = Y X ′ , with ω(X) ≥ ω(Y ) (cf Chapter I). Show that there exists a formal series U such that X = Y U , X ′ = U Y ′ . b) Let S be a formal series and let C be its centralizer, that is C = {T ∈ KhhAii | ST = T S}. Show that if T1 , T2 ∈ C and ω(T2 ) ≥ ω(T1 ), then there exists T ∈ C such that T2 = T1 T . (Hint : one may suppose ω(S) ≥ 1; let n be such that ω(S n ) ≥ ω(T1 ), ω(T2 ): use a) three times.) Let T ∈ C such that ω(T ) ≥ 1 is minimum. Show that C = K[[T ]], that is o nX an T n | an ∈ K C= n∈N

( (see Cohn 1961). 2.1 Show that for n ≥ k ≥ 1 the continuant polynomials satisfy the identities p(a1 , . . . , an )p(an−1 , . . . , ak ) − p(a1 , . . . , an−1 )p(an , . . . , ak ) = (−1)n+k p(a1 , . . . , ak−2 )

with the conventions: p(a1 , . . . , ak−2 ) = 0 if k = 1, = 1 if k = 2, and p(an−1 , . . . , ak ) = 1 if k = n. Show that the number of words in the support of p(a1 , . . . , an ) is the nth Fibonacci number Fn (F0 = F1 = 1, Fn+2 = Fn+1 + Fn ). 2.2 Show that if a1 , . . . , an are commutative polynomials, then 1

a1 +

=

1

a2 + a3 +

1 ···+

1 an

p(a1 , . . . , an ) . p(a2 , . . . , an )

114

Chapter VII. Noncommutative polynomials

2.3 Show that the entries of the matrix      a1 1 a2 1 a 1 ··· n 1 0 1 0 1 0 may be expressed by means of continuant polynomials. 3.1 Let M be an n by n polynomial matrix such that M = M1 M2 with M1 ∈ KhhAiin×p and M2 ∈ KhhAiip×n . Show that then one may choose M1 , M2 to be polynomial matrices (use the inertia theorem; see Cohn 1985).

Notes to Chapter VII Most of the results of this chapter are due to P. M. Cohn. We have already seen a result concerning noncommutative polynomials in Chapter II (Corollary II.3.3): in P. M. Cohn’s terminology, it means that KhAi is a fir (“free ideal ring”). The terminology “continuant” stems from its relation to continuous fractions (see Exercises 2.2 and 2.3). Corollary 3.2 is a special case of a more general result, stating that every polynomial matrix which is singular over the free field is not full (see Cohn 1961).

Chapter VIII

Codes and Formal Series The aim of this chapter is to present an application of formal series to the theory of (variable-length) codes. The main result (Theorem 4.1) states that every finite complete code admits a factorization into three polynomials which reflect its combinatorial structure. The first section contains some basic facts on codes and prefix codes. These are easily expressed by means of formal power series. Section 2 is devoted to complete codes and their relations to Bernoulli morphisms (Theorem 2.4). Concerning the degree of a code, we give in Section 3 only the very basic results needed in Section 4. This last section is devoted to the proof of the main result. It uses the material of the previous section and from Chapter VII.

1

Codes

Definition A code is a subset C of A∗ such that whenever u1 , . . . , un , v1 , . . . , vp in C satisfy u1 · · · un = v1 · · · vp ,

(1.1)

then n = p and ui = vi for i = 1, . . . , n. In this case, any word in C ∗ (= the submonoid generated by C) is called a message. Note that if C is a code, then C ⊂ X + (= X ∗ \ 1). Example 1.1 The set {a, ab, ba} is not a code, because the word aba has two factorizations in it: aba = a(ba) = (ab)a . Example 1.2 The set {a, ab, bb} is a code; indeed, no word in it is a prefix of another, so in each relation of the form (1.1), either u1 is a prefix of v1 or vice versa, so one has u1 = v1 and one concludes by induction on n. Example 1.3 The set {b, ab, a2 b, a3 b, . . . , an b, . . .} = a∗ b is a code, for the same reason as in Example 1.2. 115

116

Chapter VIII. Codes and Formal Series

Example 1.4 The set {a3 , a2 ba, a2 b2 , ab, ba2 , baba, bab2, b2 a, b3 } is a code, for the same reason; note that in this case, moreover no word is a suffix of another. Example 1.5 The set C = {a2 , ab, a2 b, ab2 , b2 } is a code. Indeed, let C denote its characteristic polynomial; then we have 1 − C = 1 − a2 − ab − a2 b − ab2 − b2

= (1 − b − a2 − ab) + (b − b2 − a2 b − ab2 )

= (1 − b − a2 − ab)(1 + b)

= ((1 − a − b) + (a − a2 − ab))(1 + b) = (1 + a)(1 − a − b)(1 + b) . Thus, in ZhhAii, we have (1 − C)−1 = (1 + b)−1 (1 − a − b)−1 (1 + a)−1 . By the results of Section I.4, for any proper formal series S, (1 − S)−1 = P n ∗ −1 = A∗ = A∗ is the sum of all words on A n≥0 S = S and (1 − a − b) (and hence, its nonzero coefficients are all equal to 1). Hence X  C n (1 + a) . A∗ = (1 + b) n≥0

P This shows that the series n≥0 C n has no coefficient ≥ 2, since otherwise A∗ would have such a coefficient. From X X X u1 · · · un Cn = n≥0

n≥0 u1 ,...,un ∈C

we obtain that no word has two distinct factorizations of the form u1 · · · un (ui ∈ C), so C is a code. Recall that for any language X, X denotes its characteristic series (considered as an element of QhhAii in the present chapter). One of the arguments of the last example may be generalized as follows. Proposition 1.1 Let C be a subset of A+ and let C be its characteristic series. Then C is a code if and only if one has in ZhhAii (1 − C)−1 = C ∗ = C ∗ .

(1.2)

Proof. The first equality is always true, as shown in Section I.4. We have X X X u1 · · · un = Cn = C∗ . n≥0 u1 ,...,un ∈C

n≥0

If C is a code, then the words u1 · · · un

(n ≥ 0, ui ∈ C)

are all distinct, so the left-hand side is equal to C ∗ . If C is not a code, then two of these words are equal, so the left-hand side is a series with at least one

117

1. Codes

coefficient ≥ 2: it cannot be equal to C ∗ , because the latter has only 0, 1 as coefficients.  The previous result provides an effective algorithm for testing whether a given rational subset of C of A+ is a code. Indeed, one has merely to check if the rational power series C ∗ − C ∗ is equal to 0; for this, apply Corollary II.3.4. However, there is a more direct algorithm. We give below, without proof, the algorithm of Sardinas and Patterson (see Lallement 1979, Berstel and Perrin 1985). Recall that for any language X and any word w, we denote by w−1 X the language w−1 X = {u ∈ A∗ | wu ∈ X} . More generally, if Y is a language, we denote by Y −1 X the language [ Y −1 X = w−1 X . w∈Y

Now let C be a subset of A+ . Define a sequence of languages Cn by C0 = C −1 C \ 1

Cn+1 = Cn−1 C ∪ C −1 Cn

(n ≥ 0) .

Then C is a code if and only if no Cn contains the empty word. If C is finite, the sequence (Cn ) is periodic (because each word in Cn is a factor of some word in C). The same is true if C is rational (see Berstel and Perrin 1985, Prop. I.3.3). Hence in these cases, we obtain an effective algorithm. Another way to express the fact that a set of words is a code is by means of the so-called unambiguous operations. Let X, Y be languages. We say that their union is unambiguous if they are disjoint languages. We say that their product is unambiguous if x, x′ ∈ X, y, y ′ ∈ Y , and xy = x′ y ′ implies x = x′ , y = y ′ . We say that the star X ∗ is unambiguous if X is a code. Proposition 1.2 Let X, Y be languages. (i) The union of X and Y is unambiguous if and only if X ∪ Y = X + Y . (ii) The product XY is unambiguous if and only if XY = X Y . (ii) If 1 ∈ / X, then the star X ∗ is unambiguous if and only if X ∗ = X ∗ . Proof. The first two assertions are a direct consequence of their definition. The last one is merely a reformulation of Proposition 1.1.  We have already met a family of codes in Section II.3: the prefix codes. A code is a prefix code if no word in it is a prefix of another word in it (this condition is sufficient to ensure codicity). Symmetrically, one defines suffix codes. A code is called bifix if it is both prefix and suffix. Proposition 1.3 Let C be a code such that for any word v in C ∗ , one has v −1 C ∗ ⊂ C ∗ . Then C is a prefix code. v

Note the converse: for any set C and for any word v in C ∗ , one has C ∗ ⊂ C ∗.

−1

118

Chapter VIII. Codes and Formal Series

Proof. Suppose u = vw, with u, v in C and w ∈ A∗ . We have to show that w = 1. Now w = v −1 u ∈ v −1 C ∗ ⊂ C ∗ , hence w ∈ C ∗ . Therefore w = c1 · · · cn (ci ∈ C) and u = vc1 · · · cn ∈ C. The only possibility for C to be a code is n = 0, that is w = 1, and C is a prefix code.  Proposition 1.4 Let C be a prefix code such that CA∗ ∩ wA∗ is nonempty for any word w. Let P be the set of proper prefixes of the words in C. Then one has in ZhhAii C − 1 = P (A − 1) . Proof. Let P ′ = A∗ \ CA∗ . Then, by Proposition II.3.1, we have A∗ = C ∗ P ′ . But, because C is a prefix code, the conditions u1 · · · un q = v1 · · · vp r, ui , vj ∈ C, q, r ∈ P ′ imply n = p, ui = vi for i = 1, . . . , n, hence also q = r. This shows that the product C ∗ P ′ is unambiguous, hence by Proposition 1.2, we have A∗ = C ∗ P ′ . Now, by Proposition 1.1, A∗ = (1 − A)−1 and C ∗ = (1 − C)−1 . Moreover, the empty word is in P ′ , so P ′ is invertible in ZhhAii. Hence 1−A = P ′−1 (1−C), which implies C − 1 = P ′ (A − 1). It remains to show that P = P ′ . Let w be in P ; then w is a proper prefix of some word in C and so has no prefix in C, C being a prefix code; hence w∈ / CA∗ =⇒ w ∈ P ′ . Let w be in P ′ . By assumption, there are words c ∈ C, u, v ∈ A∗ such that cu = wv; as w ∈ / CA∗ , w must be a proper prefix of c, so w ∈ P .  Let C be a code. Define, for any word u, the series Su inductively by S1 = 1 Su = a−1 Sv + (Sv , 1)a−1 C ,

for u = va (a ∈ A)

Note that, obviously, Su has nonnegative coefficients. The reader may verify that the support of Su consists of proper suffixes of C (cf. Exercise 1.3). Lemma 1.5 Let C be a code. Then for any word u, u−1 (C ∗ ) = Su C ∗ . In particular, Su is a characteristic series. If C is finite, then Su is a polynomial. Proof. We shall use the formulas of Lemma I.6.2. We prove u−1 (C ∗ ) = Su C ∗ by induction on |u|. If u = 1, it is clearly true. Let u = va, (a ∈ A). Then by induction v −1 (C ∗ ) = Sv C ∗ . Thus, by Lemma I.6.2, u−1 (C ∗ ) = a−1 v −1 (C ∗ ) = (a−1 Sv )C ∗ + (Sv , 1)(a−1 C ∗ ) = (a−1 Sv )C ∗ + (Sv , 1)(a−1 C)C ∗ = Su C ∗ . Now, since u−1 (C ∗ ) is obviously a characteristic series, the same holds for Su . It is easily verified by induction that Su is a polynomial if C is finite.  One defines symmetrically the series Pu ∈ ZhhAii by P1 = 1 Pav = Pv a−1 + (Pv , 1)Ca−1 ,

for a ∈ A and v ∈ A∗

119

2. Completeness

Now we define, for a couple (u, v) of words another series in the following way: Fu,1 = 0 Fu,av = (Pv , 1)Su a−1 + Fu,v a−1 . As above, the series Fu,v clearly has nonnegative coefficients. Proposition 1.6 Let C be a code. Then for any words u and v, u−1 (C ∗ )v −1 = Su C ∗ Pv + Fu,v . In particular, Fu,v is a characteristic series. If C is finite, then Fu,v is a polynomial. Proof (Induction on |v|). The result is obvious if v = 1 by Lemma 1.5. Let a ∈ A. Then u−1 (C ∗ )(av)−1 = [u−1 (C ∗ )v −1 ]a−1 is equal, by induction and Lemma I.6.2, to (Su C ∗ Pv )a−1 + Fu,v a−1 = Su C ∗ (Pv a−1 ) + (Pv , 1)Su (C ∗ a−1 ) + (Pv , 1)Su a−1 + Fu,v a−1 = Su C ∗ (Pv a−1 ) + (Pv , 1)Su C ∗ (Ca−1 ) + Fu,av = Su C ∗ Pav + Fu,av . This proves the formula. Now, since Su C ∗ Pv has nonnegative coefficients and since u−1 (C ∗ )v −1 is a characteristic series, the same holds for Fu,v . If C is finite, it is easily seen by induction on the definition that Fu,v is a polynomial. 

2

Completeness

Definition A language C ⊂ A∗ is complete if, for any word w, the set C ∗ ∩ A∗ wA∗ is nonempty. Lemma 2.1 If C is complete, then any word w is either a factor of a word in C or may be written as w = smp , with m ∈ C ∗ and where s (p) is a suffix (prefix) of a word of C. Proof. We have xwy ∈ C ∗ for some words x, y. Let us represent a word in C ∗ schematically by

Then we have two cases: 1)

w

120

Chapter VIII. Codes and Formal Series

2)

w In the first case, w is a factor of a word in C. In the second case, w = smp as in the lemma.  Definition A Bernoulli morphism is a mapping π : A∗ → R such that (i) (ii) (iii) (iv)

π(w) > 0 for any word w, π(1) = 1, π(uv) P = π(u)π(v) for any words u, v, π(a) = 1. a∈A

It is called uniform if π(a) = 1/|A| for any letter a. We define for any language X the measure of X by X π(X) = π(w) w∈X

(it may be infinite). We shall frequently use the following inequalities: X π(∪Xi ) ≤ π(Xi ) π(XY ) ≤ π(X)π(Y ) .

Note that, for any n, one has π(An ) = 1. Lemma 2.2 Let C be a code. Then π(C) ≤ 1. Proof. Since C is the limit of its finite subsets, it is enough to show the lemma in the case where C is finite. Let p be the maximal length of words in C. Then C n ⊂ A ∪ A2 ∪ · · · ∪ Apn . Thus π(C n ) ≤ pn. Now, as C is a code, each word in C n has only one factorization of the form u1 · · · un (ui ∈ C). As π is multiplicative, we obtain π(C n ) = π(C)n . Hence π(C)n ≤ pn . This shows that π(C) ≤ 1.  Lemma 2.3 Let C be a finite complete language. Then π(C) ≥ 1. Proof. By Lemma 2.1, we may write A∗ = SC ∗ P ∪ F , where S, P, F are finite languages. Thus ∞ = π(A∗ ) ≤ π(S)π(C ∗ )π(P ) + π(F ) .

121

2. Completeness This shows that π(C ∗ ) = ∞. Now [ C∗ = Cn n≥0

P so that π(C ∗ )P ≤ n≥0 π(C n ). Moreover, π(C n ) ≤ π(C)n , π being multiplicative. So ∞ ≤ n≥0 π(C)n , which shows that π(C) ≥ 1. 

Theorem 2.4 (Sch¨ utzenberger and Marcus 1959, Bo¨e et al. 1980) Let C be a finite subset of A∗ and let π be a Bernoulli morphism. Then any two of the following assertions imply the third one: (i) C is a code, (ii) C is complete, (iii) π(C) = 1 . Note that this gives an algorithm for testing whether a given finite code is complete (see Exercise 2.3). We need another lemma. Lemma 2.5 Let X be a language and let w be a word such that X ∩ A∗ wA∗ is empty. Then π(X) < ∞. Proof. Let ℓ = |w| and for i = 0, . . . , ℓ − 1 Xi = {v ∈ X | |v| ≡ i mod ℓ} . Then Xi ⊂ Ai (Aℓ \ w)∗ . Indeed v ∈ Xi implies v = uv1 · · · vn with |u| = i and for any j, |vj | = ℓ; by assumption, w is not factor of v, hence w is none of the vj ’s: thus vj ∈ Aℓ \ w, which proves the claim. Now π(Aℓ \ w) = π(Aℓ ) − π(w) = 1 − π(w) < 1 and π[(Aℓ \ w)∗ ] = π ≤

X

ℓ

[

 X (Aℓ \ w)n ≤ π[(Aℓ \ w)n ]

n≥0

n

n≥0

[π(A \ w)] < ∞ .

n≥0

Thus π(Xi ) = π[Ai (Aℓ \ w)∗ ] ≤ π(Ai )π[(Aℓ \ w)∗ ] < ∞ and since X = ∪0≤i≤ℓ−1 Xi , we obtain π(X) < ∞.  Proof of Theorem 2.4. Lemma 2.2 and 2.3 show that (i) and (ii) imply (iii). Let C be a code with π(C) = 1. Suppose C is not complete. Then for some word w, C ∗ ∩ A∗ wA∗ is empty. P Hence, by Lemma 2.5, π(C ∗ ) < ∞. As C is a ∗ code, π(C ) is equal to the sum n≥0 π(C)n . The latter being finite, we deduce that π(C) < 1, a contradiction. Let C be complete and π(C) = 1. Then C n is complete for any n; indeed, for any word w, there are words u, v, c1 , . . . , cp (ci ∈ C) such that uwv = c1 · · · cp (C being complete). Let r be such that p + r is a multiple of n; then uwvcr1 = c1 · · · cp cr1 ∈ (C n )∗ , which shows that (C n )∗ ∩ A∗ wA∗ is not empty. Hence

122

Chapter VIII. Codes and Formal Series

C n is complete. Thus, by Lemma 2.3, π(C n ) ≥ 1 for any n. But as usually π(C n ) ≤ π(C)n = 1, thus π(C n ) = π(C)n for any n. Suppose C is not a code. Then for some words u1 , . . . , un , v1 . . . , vp in C we have u1 · · · un = v1 · · · vp and u1 6= v1 . Hence u1 · · · un v1 · · · vp = v1 · · · vp u1 · · · un , and we have obtained a word in C n+p which has two distinct factorizations. Hence
π(C n+p ) = π {w1 · · · wn+p | wi ∈ C} X < π(w1 · · · wn+p ) = π(C n+p ) w1 ,...,wn+p ∈C

which is a contradiction.  Let π be a Bernoulli morphism. Since π is multiplicative, it may be extended to an algebra morphism, still denoted by π, π : ZhAi → R by the formula X  X π (P, w)w = (P, w)π(w) . w

w

Note that, because the measure of A is 1, one has π(A − 1) = 0 . Theorem 2.6 (Sch¨ utzenberger 1965) Let C be a finite code such that for any word w, the set C ∗ ∩ wA∗ is nonempty. Then C is a prefix code. Proof. Let C ′ be the set of words in C having no proper prefix in C, that is C ′ = C \ CA+ . Clearly C ′ is a prefix code. Moreover, if w is a word, then for some words c1 , . . . , cn ∈ C, u ∈ A∗ , one has by assumption c1 · · · cn = wu . Then either c1 ∈ C ′ , or c1 has a prefix in C ′ . Thus C ′ A∗ ∩ wA∗ is nonempty. Let P be the set of proper prefixes of the words in C ′ . Then by Proposition 1.4, C ′ − 1 = P (A − 1). Apply the morphism π : ZhAi → R, obtaining π(C ′ − 1) = 0 because π(A − 1) = 0. Thus π(C ′ ) = 1. As C is a code, we have by Lemma 2.2, π(C) ≤ 1. But C ′ ⊂ C and π is positive. Hence C = C ′ is prefix.  Theorem 2.7 (Reutenauer 1985) Let P in NhAi be without constant term such that P − 1 = X(A − 1)Y for some polynomials X, Y in RhhAii. Then P = C for some finite complete code C. Furthermore, if Y ∈ R (X ∈ R), then C is a prefix (suffix) code. Proof. 1. Note that if S, T are formal series, then supp(ST ) ⊂ supp(S) supp(T ) .

3. The Degree of a Code

123

Moreover, if S is proper, then supp(S ∗ ) ⊂ supp(S)∗ . 2. We have 1 − P = X(1 − A)Y . By assumption, 1 − P is invertible in RhhAii. The same holds for 1 − A since its inverse is A∗ = A∗ . This shows that X and Y are also invertible. So we obtain (1 − P )−1 = Y −1 (1 − A)−1 X −1 which implies (1 − A)−1 = Y (1 − P )−1 X . Thus A∗ = Y P ∗ X .

(2.1)

By 1, this implies that each word w may be written as w = ymx, with y ∈ supp(Y ), m ∈ supp(P )∗ and x ∈ supp(X). Let C = supp(P ) and let u be a word such that |u| > deg(X), deg(Y ). Let v be any word. Then w = uvu may be written uvu = ymx as above, which shows, by the choice of u, that m = v1 vv2 . Hence C ∗ ∩ A∗ vA∗ is nonempty: we have shown that C is complete. Thus, by Lemma 2.3, π(C) ≥ 1 (where π is some Bernoulli morphism). Now, as P − 1 = X(A − 1)Y , we obtain π(P ) = 1. Hence 1 ≤ π(C) ≤ π(P ) = 1 because P has nonnegative integer coefficients. This shows, π being positive, that P = C and that π(C) = 1. Hence, by Theorem 2.4, C is a code, and thus a finite complete code. Suppose now that Y ∈ R. Then, as above, Eq. (2.1) shows that for any word v, one has vu = mx for some words m ∈ C ∗ , x ∈ supp(X) (u being chosen as before). Then, as |u| > |x|, we obtain m = vv1 which shows that C ∗ ∩ vA∗ is nonempty. We conclude by Theorem 2.6. 

3

The Degree of a Code

Given a monoid M , recall that an ideal in M is a nonempty subset J which is closed for left and right multiplication by elements of M . Moreover, an idempotent is an element e which is equal to its square, that is e2 = e. Theorem 3.1 (Suschkewitsch 1928) Let M be a finite monoid. There exists in M an ideal J which is contained in any ideal of M . Let e be an idempotent in J. Then eM e is a finite group whose neutral element is e. This ideal will be called the minimal ideal of M Proof. 1. Let J be the intersection of all ideals in M . Clearly J is closed for multiplication by elements of M . We have only to verify that it is not empty. But let m be the product of all elements of M , in some order. Then m is in each ideal of M , and hence in J.

124

Chapter VIII. Codes and Formal Series

2. We use the following classical fact: if a ∈ M , then some positive power of a is an idempotent. Indeed, chose i, j ≥ 1 such that j ≥ i and that ai = ai+j (this is possible because the set {a, a2 , . . . , an , . . .} is finite). Let k = j − i. Then ai+k is idempotent because ai+k ai+k = ak ai+i+k = ak ai+j = ak ai = ak+i . 3. Clearly, eeme = eme = emee and emeem′ e = e(mem′ )e, hence eM e is a (finite) monoid whose neutral element is M . 4. Let a = eme be in eM e. We show the existence of b ∈ eM e such that ab = e. We have a = et for some t ∈ M . Now M aM is an ideal of M contained in J (because M aM = M etM , e ∈ J and J is an ideal), hence M aM = J (J being minimal). Thus e = uav for some elements u, v of M . Next, e = uetv = uuetvtv = un e(tv)n for any n ≥ 1. Choose n such that (tv)n is idempotent. Then e = un e(tv)n = un e(tv)n (tv)n = e(tv)n = etv(tv)n−1 = aw (recall that et = a). But a = eme implies ae = eme2 = eme = a, whence e = aw = aew and e = e2 = aewe. Let b = ewe ∈ eM e. Then e = ab. 5. Symmetrically, we have e = ca for some c in eM e. Then, classically c = ce = cab = eb = b. This shows that each element of eM e has an inverse in eM e, that is, eM e is a group.  Theorem 3.2 Let C be a finite complete code. There exist a finite monoid M and a surjective morphism φ : A∗ → M such that C ∗ = φ−1 φ(C ∗ ). Let J be the minimal ideal of M . There exists an idempotent e in J ∩ φ(C ∗ ); further φ(C ∗ ) ∩ eM e is a subgroup of the group eM e. It will not be shown here that the index of φ(C ∗ ) ∩ eM e in eM e depends only on C; for this, we refer the reader to the book by Berstel and Perrin (1985). This being admitted, we introduce the following definition. Definition With the notation of Theorem 3.2, the index of eM e ∩ φ(C ∗ ) in eM e is called the degree of C. Proof of Theorem 3.2. Clearly, C ∗ is a rational subset of A∗ (cf. Section III.1). Hence, by Kleene’s theorem (Theorem III.1.1), it is recognizable. This shows that there exist a finite monoid M , a monoid morphism φ : A∗ → M , and a subset N of M such that C ∗ = φ−1 (N ). Clearly, we may assume that φ is surjective; then N = φ(C ∗ ) and C ∗ = φ−1 φ(C ∗ ). Let J be the minimal ideal of M and w a word in φ−1 (J). Then C ∗ ∩A∗ wA∗ is nonempty (because C is complete), hence there exist words u, v such that uwv is in C ∗ . Now m = φ(uwv) is in φ(C ∗ ) and also in J (because m = φ(u)φ(w)φ(v), φ(w) ∈ J, and J is an ideal). Some power e = mn with n ≥ 1 of m is idempotent and still lies in φ(C ∗ ) ∩ J. Now, φ(C ∗ ) is clearly a submonoid of M . Hence, the product of any two elements of eM e ∩ φ(C ∗ ) lies in eM e ∩ φ(C ∗ ). Take a ∈ eM e ∩ φ(C ∗ ). Then for some n ≥ 2, an = e (eM e being a finite group). Then an−1 is the inverse of a in eM e, and belongs to eM e ∩ φ(C ∗ ). Thus, the latter is a subgroup of eM e. 

4

Factorization

Theorem 4.1 (Reutenauer 1985) Let C be a finite complete code. Then there exist polynomials X, Y, Z in ZhAi such that C − 1 = X(d(A − 1) + (A − 1)Z(A − 1))Y

(4.1)

4. Factorization

125

and (i) d is the degree of C, (ii) C is prefix (suffix) if and only if Y = 1 (X = 1). Example 4.1 We have a2 + a2 b + ab + ab2 + b2 − 1 = (1 + a)(a + b − 1)(1 + b) . The corresponding code is neither prefix nor suffix, but synchronizing (that is of degree 1). Example 4.2 Let C be the square of the code of Example 4.1. Then C is of degree 2 and C − 1 = (1 + a)(2(a + b − 1) + (a + b − 1)(1 + b)(1 + a)(a + b − 1))(1 + b) . Example 4.3 We have a3 + a2 ba + a2 b2 + ab + ba2 + baba + bab2 + b2 a + b3 − 1

= 3(a + b − 1) + (a + b − 1)(2 + a + b + ab)(a + b − 1) . The corresponding code is a bifix code and has degree 3.

The following corollary (which also uses Theorem 2.7) characterizes completely finite complete codes. Corollary 4.2 (Reutenauer 1985) Let C be a language not containing the empty word. Then the following conditions are equivalent: (i) C is a complete finite code. (ii) There exist polynomials P, S in ZhAi such that C − 1 = P (A − 1)S .



In order to prove Theorem 4.1, we need the following lemma. Lemma 4.3 Let C be a finite complete code of degree d. Then there exist words u1 , . . . ud , v1 , . . . , vd , with u1 , v1 ∈ C ∗ , such that for any i, 1 ≤ i ≤ d: X ∗ −1 A∗ = u−1 i (C )vj 1≤j≤d

and for any j, 1 ≤ j ≤ d: X ∗ −1 u−1 A∗ = i (C )vj . 1≤i≤d

Proof. By Theorem 3.2 there exist a finite monoid M and a surjective morphism φ : A∗ → M such that C ∗ = φ−1 φ(C ∗ ); moreover, there exists an idempotent e in J ∩ φ(C ∗ ), where J is the minimal ideal of M , G = eM e is a finite group and H = eM e ∩ φ(C ∗ ) is a subgroup of G of index d.

126

Chapter VIII. Codes and Formal Series

Let u1 , . . . ud , v1 , . . . , vd be words in φ−1 (G) such that [ G= φ(vi )H

(4.2)

1≤i≤d

and G=

[

Hφ(uj )

1≤j≤d

(disjoint unions). By elementary group theory, we may assume that φ(u1 ) = φ(v1 ) = e (hence u1 , v1 ∈ φ−1 (e) ⊂ φ−1 φ(C ∗ ) = C ∗ ) and that φ(ui ) is the inverse of φ(vi ) in G. Let 1 ≤ j ≤ d and w be a word. Then there exists one and only one i, ∗ −1 ∗ 1 ≤ i ≤ d, such that w ∈ u−1 i (C )vj , that is ui wvj ∈ C . Indeed, the element eφ(wvj ) of G is in some φ(vi )H by Eq. (4.2). Hence, φ(ui wvj ) = φ(ui )eφ(wvj ) ∈ φ(ui )φ(vi )H = eH = H, which implies that ui wvj ∈ φ−1 (H) ⊂ φ−1 φ(C ∗ ) = C ∗ . Conversely, ui wvj ∈ C ∗ implies φ(ui wvj ) ∈ eM e ∩ φ(C ∗ ) = H, because φ(ui wvj ) = eφ(ui wvj )e is already in eM e. Hence eφ(wvj ) = φ(vi )φ(ui wvj ) ∈ φ(vi )H, and i is completely determined by j and w. We have shown that one has the disjoint union, for any j, 1 ≤ j ≤ d: [ ∗ −1 A∗ = u−1 i (C )vj . 1≤i≤d

But this is equivalent to the last relation of the lemma. By symmetry, we have also the first.  We easily derive the following lemma Lemma 4.4 Let C be a finite complete code of degree d. Then there exist polynomials P, P1 , S, S1 , Q, G1 , D1 with coefficients 0, 1 such that (i) (ii) (iii) (iv) (v) (vi)

dA∗ − Q = SC ∗ P . A∗ − G1 = SC ∗ P1 . A ∗ − D 1 = S1 C ∗ P . P1 , S1 have constant term 1. G1 , D1 have constant term 0. If C is a prefix (suffix) code, then S1 = 1 (P1 = 1).

Proof. We use Lemma 4.3 and the notation of Section 1. We have, by Proposi∗ −1 tion 1.6, u−1 = Sui C ∗ Pvj + Fui ,vj ; moreover, by Lemma 1.5 and Propoi (C )vj sition 1.6, Sui , Pvj and Fui ,vj are polynomials with nonnegative coefficients. Now, by Lemma 4.3, for any i X X Sui C ∗ Pvj + Fui ,vj A∗ = 1≤j≤d

1≤j≤d

and for any j A∗ =

X

1≤i≤d

Sui C ∗ Pvj +

X

1≤i≤d

Fui ,vj

127

4. Factorization Let P =

X

Pvj ,

S=

1≤j≤d

G1 =

X

X

Sui ,

P1 = Pv1 ,

S1 = Su1

1≤i≤d

Fui ,v1 ,

D1 =

X

Fu1 ,vj

Q=

j

i

X

Fui ,vj .

i,j

Then we obtain dA∗ = SC ∗ P + Q,

A∗ = SC ∗ P1 + G1 ,

A∗ = S1 C ∗ P + D1 ,

(4.3)

which proves (i), (ii) and (iii). ∗ −1 ∗ As u1 ∈ C ∗ by Lemma 4.3, u−1 1 (C ) contains 1, hence u1 (C ) has constant ∗ ∗ −1 term 1. As u1 (C ) = Su1 C by Lemma 1.5, S1 = Su1 must have constant term 1. TheP same holds for P1 by symmetry, and proves (iv). As S = Sui , the Sui ’s are nonnegative and as Su1 has constant term 1, i

S has nonnegative constant term. Moreover, P1 has constant term 1. Hence, because A∗ has constant term 1 and by Eq. (4.3), G1 has constant term 0. Similarly, D1 has constant term 0. This proves (v). ∗ ∗ Suppose now that C is prefix. Then, by Proposition 1.3, u−1 1 (C ) = C ∗ ∗ ∗ ∗ −1 −1 ∗ (because u1 ∈ C ). Hence u1 (C ) = C . As by Lemma 1.5, u1 (C ) = Su1 C , we obtain S1 = Su1 = 1. Similarly, if C is suffix, then P1 = 1. This proves (vi).  Given a Bernoulli morphism π, define a mapping λ for each word w by λ(w) = π(w) |w| . For each language X, define λ(X) by X λ(X) = λ(w) ∈ R+ ∪ ∞ . w∈X

This is called the average length of X. On the other hand λ extends to a linear mapping ZhAi → R by X λ(P ) = (P, w)λ(w) . w

Lemma 4.5 Let P1 , . . . , Pn be polynomials. Then X λ(P1 · · · Pn ) = π(P1 ) · · · π(Pi−1 )λ(Pi )π(Pi+1 ) · · · π(Pn ) . 1≤i≤n

Proof. For n = 2, it is enough, by linearity, to prove the lemma when P1 = u, P2 = v are words. But in this case λ(uv) = π(uv) |uv| = π(u)π(v)(|u| + |v|) = π(u)|u|π(v) + π(u)π(v)|v| = λ(u)π(v) + π(u)λ(v) . The general case is easily proved by induction. 

128

Chapter VIII. Codes and Formal Series

Proof of Theorem 4.1. 1. First, note that the “if” part of (ii) is a consequence of Theorem 2.7. We use the notation of Lemma 4.4. We have A∗ − G1 = (1 − A)−1 − G1 = (1 − A)−1 (1 − (1 − A)G1 ). As A∗ − G1 = SC ∗ P1 and P1 has constant term 1 (Lemma 4.4), P1 is invertible in ZhAi and we obtain from SC ∗ P1 = (1 − A)−1 (1 − (1 − A)G1 ) , by multiplying by 1 − A on the left and by P1−1 on the right, (1 − A)SC ∗ = (1 − (1 − A)G1 )P1−1 .

(4.4)

Multiply the relation (i) of Lemma 4.4 by 1 − A on the left. This yields d − (1 − A)Q = (1 − A)SC ∗ P . Hence, by Eq. (4.4), d − (1 − A)Q = (1 − (1 − A)G1 )P1−1 P . Note that, because G1 has no constant term, 1−(1−A)G1 is invertible in ZhhAii, so that we obtain, by multiplying the previous relation by P1 (1 − (1 − A)G1 )−1 on the left P = P1 (1 − (1 − A)G1 )−1 (d − (1 − A)Q) . 2. We apply Corollary VII.4.3 to the last equality: there exist E, F, G, H in ZhAi such that P1 = EF,

1 − (1 − A)G1 = GF

d − (1 − A)Q = GH,

P = EH

(4.5)

By Proposition VII.4.4 applied to the second equality (with 1 − A instead of Y ), we obtain G ≡ ±1 mod (1 − A)ZhAi . Replacing if necessary E, F, G, H by their opposites, we may suppose that G ≡ +1, and hence we obtain, again by Proposition VII.4.4, and by the third equality in Eq. (4.5), that H ≡ d mod (1 − A)ZhAi, which implies P = E(d + (A − 1)R) ,

R ∈ ZhAi .

(4.6)

3. We have A∗ − D1 = (1 − A)−1 (1 − (1 − A)D1 ) so that by Lemma 4.4 (iii), S1 C ∗ P = (1 − A)−1 (1 − (1 − A)D1 ) . As D1 has constant term 0, (1 − (1 − A)D1 ) is invertible in ZhhAii; moreover S1 is also invertible because it has constant term 1. So we obtain, by multiplying by (1 − C)S1−1 on the left and by (1 − (1 − A)D1 )−1 (1 − A) on the right, (1 − C)S1−1 = P (1 − (1 − A)D1 )−1 (1 − A) . Now we use Eq. (4.6) and multiply by −S1 on the right, thus obtaining C − 1 = E(d + (A − 1)R)(1 − (1 − A)D1 )−1 (A − 1)S1 .

129

4. Factorization 4. By Corollary VII.4.3, there exist E ′ , F ′ , G′ , H ′ ∈ ZhAi such that E(d + (A − 1)R) = E ′ F ′ , 1 − (1 − A)D1 = G′ F ′ (A − 1)S1 = G′ H ′ , C − 1 = E ′ H ′ .

(4.7)

Let π be any Bernoulli morphism. Replacing if necessary E ′ , F ′ , G′ , H ′ by their opposites, we may assume that π(F ′ ) ≥ 0 . So, by Eq. (4.7) and Proposition VII.4.4, we obtain (since π(A − 1) = 0) G′ = 1 + (A − 1)G′′ ,

F ′ = 1 + (A − 1)F ′′

(4.8)

for some G′′ , F ′′ ∈ ZhAi. This and Eq. (4.7) imply that (A − 1)S1 = (1 + (A − 1)G′′ )H ′ = H ′ + (A − 1)G′′ H ′ . Thus, we have H ′ = (A − 1)H ′′ ,

H ′′ ∈ ZhAi .

(4.9)

Now, Eqs. (4.7) and (4.8) imply also E(d + (A − 1)R) = E ′ (1 + (A − 1)F ′′ ) . 5. We now apply Theorem VII.2.2 to this equality and denote by pi the continuant polynomial p(a1 , . . . , ai ) and p˜i = p(ai , . . . , a1 ). Thus there exist polynomials U, V ∈ QhAi such that E = U pn , ′

d + (A − 1)R = p˜n−1 V,

E = U pn−1 ,

1 + (A − 1)F ′′ = p˜n V .

(4.10)

Applying Corollary VII.1.3 to the second and the last equalities (with X → p˜n−1 or p˜n , Y → A − 1, Q1 → 0, P → V , R → d or 1), we obtain that the left Euclidean division of p˜n−1 and p˜n by A− 1 is possible, that is p˜n−1 and p˜n are both congruent to a scalar mod(A − 1)QhAi. This implies, by Proposition VII.2.3, that pn−1 and p˜n−1 (pn and p˜n ) are congruent to the same scalar mod (A − 1)QhAi. lary VII.4.2, they have the same content c(pn−1 ) = c(˜ pn−1 ),

c(pn ) = c(˜ pn ) .

(4.11) Moreover, by Corol(4.12)

6. As D1 has coefficients 0, 1, the polynomial 1 − (A − 1)D1 is primitive. Hence, by Eq. (4.7) and by Gauss’s Lemma, G′ and F ′ are primitive. As by Eqs. (4.10) and (4.8) p˜n V = 1 + (A − 1)F ′′ = F ′ , we obtain by Gauss’s Lemma c(˜ pn )c(V ) = 1

130

Chapter VIII. Codes and Formal Series

and p¯ ˜n V = F ′ . Hence, by Proposition VII.4.4 and Eq. (4.8), V = ε + (A − 1)V ′ ,

ε = ±1, V ′ ∈ ZhAi .

(4.13)

Furthermore, C − 1 is primitive, hence so is E ′ by Eq. (4.7). As E ′ F ′ = E(d + (A − 1)R) by Eq. (4.7) and E ′ , F ′ are primitive, we obtain by Gauss’s Lemma that d + (A − 1)R is primitive. Thus by Eq. (4.10) and Gauss’s Lemma again d + (A − 1)R = p¯˜n−1 V . This implies, by Proposition VII.4.4 and Eq. (4.13), p¯ ˜n−1 = εd + (A − 1)L,

L ∈ ZhAi .

By Eqs. (4.11) and (4.12), we obtain that p¯n−1 and p¯˜n−1 are congruent to the same scalar mod(A − 1)QhAi. Hence p¯n−1 = εd + (A − 1)M with M ∈ QhAi. But p¯n−1 − εd = (A − 1)M and A − 1 is primitive, so that c(M ) = c(¯ pn−1 − εd) ∈ N and M ∈ ZhAi, by Eq. (4.2) in Chapter VII. We have seen that E ′ is primitive, so that by Gauss’s Lemma and Eq. (4.10), we have E ′ = U p¯n−1 which implies E ′ = U (εd + (A − 1)M ) . Hence, by Eqs. (4.7) and (4.9), C − 1 = U (εd + (A − 1)M )(A − 1)H ′′ , where all polynomials are in ZhAi and where ε = ±1. This shows that we have a relation of the form C − 1 = X(ε′ d + (A − 1)D)(A − 1)Y , where X = ±U , Y = ±H ′′ , ε′ d + (A − 1)D = ±(εd + (A − 1)M ) are chosen in such a way that, for some Bernoulli morphism π, one has π(X) ≥ 0, π(Y ) ≥ 0 . 7. Apply Lemma 4.5 to this relation, using the fact that π(A −1) = 0; we obtain λ(C − 1) = π(X)ε′ dλ(A − 1)π(Y ) .

4. Factorization

131

Now λ(1) = 0, λ(C) > 0, λ(A) > 0, and we obtain ε′ dπ(X)π(Y ) > 0 . This shows that ε′ = 1 and proves Eq. (4.1) and (i). 8. Now, if C is a prefix code, we have by Lemma 4.4 (vi) that S1 = 1. Hence, by Eq. (4.7), A − 1 = G′ H ′ , which implies by Eq. (4.9) A − 1 = G′ (A − 1)H ′′ . Hence H ′′ = ∓1, and we obtain Y = ±1. But π(Y ) ≥ 0, so Y = 1. On the other hand, if C is suffix, then P1 = 1 by Lemma 4.4 (vi). Then, by Eq. (4.5), E = ±1 which implies by Eq. (4.10) and Gauss’s Lemma that U = ±1. Thus X = ±1. As π(X) ≥ 0, we obtain X = 1. This proves the theorem. 

Exercises for Chapter VIII 1.1 Show that a submonoid of A∗ is of the form C ∗ , C a code, if and only if it is free (that is isomorphic to some free monoid). Show that a submonoid M of A∗ is free if and only if for any words u, v, w u, uv, vw, w ∈ M =⇒ v ∈ M . 1.2 Show that, given rational languages K, L, it is decidable whether their union (their product, the star of K) is unambiguous. 1.3 Show that Su (Pu , Fu,v ) as defined in Section 1 is a sum of proper suffixes (prefixes, factors) of words of C. 2.1 Show that for a finite code C the three following conditions are equivalent: (i) C is a complete and prefix code. (ii) For any word w, wA∗ ∩ CA∗ is not empty. (iii) For any word w, wA∗ ∩ C ∗ is not empty. 2.2 Let C be a finite complete language. Show that for any word w, there exists some power of a conjugate of w which is in C ∗ (two words w, w′ are conjugate if w = uv, w′ = vu for some words u, v). 2.3 Deduce from Theorem 2.4 an algorithm to show that a finite set C is a code (hint: it is decidable whether C is complete, since the set of factors of a rational language is rational). 3.1 Show that if e, e′ are idempotents in the minimal ideal J of a finite monoid M , then there exists an idempotent e1 in J which is a right multiple of E and a left multiple of e′ . Show that the mapping a 7→ ae1 defines a group isomorphism eM e → e1 M e1 . Deduce that all the maximal groups in J are isomorphic. 3.2 Let C be a finite complete code. Show that C is synchronizing (that is of degree 1) if and only if for some word w, one has wA∗ w ⊂ C ∗ . 4.1 Let C be a finite complete code which is bifix. Let n be such that an ∈ C for some letter a. a) Show that for any i, 1 ≤ i ≤ n, Ci = a−i C is a prefix set such that Ci A∗ ∩ wA∗ is not empty for any word w.

132

Chapter VIII. Codes and Formal Series b) Show that the set of proper suffixes of C is the disjoint union of the Ci ’s. c) Deduce that C i − 1 = Pi (A − 1) and that n  X Pi (A − 1) . C − 1 = n(A − 1) + (A − 1) i=1

Show that n is the degree of C. Show that it is also equal to the average length of C (cf. Perrin 1977).

Notes to Chapter VIII Theorem 4.1 is a non commutative generalization of a theorem due Sch¨ utzenberger (1965). Corollary 4.2 is a partial answer to the main conjecture in the theory of finite codes, the factorization conjecture which states that P and S may be chosen to have nonnegative coefficients (or equivalently coefficients 0 and 1). Finite complete codes are maximal codes, and conversely, every maximal code is complete. Most of the general results on codes are stated here in the finite case. However, they hold for rational and even for thin codes. For a general exposition of the theory of codes, see the book by Berstel and Perrin (1985). Another illustration of the close relation between codes and formal series is the following result (roughly): a thin code is bifix if and only if its syntactic algebra is semisimple (Reutenauer 1981, Berstel and Perrin 1985).

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138

Index

Index adjoint morphism, 24 algebra group –, 54 Hadamard –, 47, 53 monoid –, 47 syntactic –, 22, 34 algorithm – of Sardinas and Patterson, 117 reduction –, 33 alphabet, 2 annihilator, 63 automaton, 24 average length, 127 Bernoulli morphism, 120 bifix code, 117 Boolean semiring, 2 cancellative – right module, 106 characteristic – series, 5 – zero, 54 code, 115 codimension, 25 coefficient, 2 commutative semiring, 1 complete – language, 119 – topological space, 4 completely integrally closed, 73 congruence monoid –, 35 semiring, 15 syntactic –, 47 conjecture, 46, 70, 132 conjugate, 131 constant term, 4, 106 content of a polynomial, 111 continuant polynomial, 102 continuous fraction, 114 degree – of a code, 124

– of a polynomial, 3, 99 – of growth, 94 denominator minimal –, 50 dense, 4 dependent, 100 dimension of a linear representation, 8 discrete topology, 3 distance ultrametric –, 3 dominating eigenvalue, 76 strictly – , 78 eigenvalue dominating –, 76 strictly dominating –, 78 eigenvalues of a rational series, 50 Eisenstein’s criterion, 18 equality set, 96 equality set, 47 Euclidean, 99 – algorithm, 99 – division, 101 exponential polynomial, 54 extension Fatou –, 83 factorization conjecture, 132 family locally finite –, 4 summable –, 4 Fatou – extension, 83 – lemma, 72 strong – ring, 88 weak – ring, 88 finitely generated – Abelian group, 57 – module, 8 fir, 114 formal series, 2 free – ideal ring, 114 – monoid, 2

139

Index full matrix, 108 Gauss’s lemma, 111 generating function, 37 geometric series, 26, 56 group algebra, 54 growth degree of –, 94 polynomial –, 87, 94 Hadamard – algebra, 53 – product, 11 Hankel – -like property, 19 – matrix, 25, 52 Hilbert’s tenth problem, 90 ideal – in a monoid, 123 minimal –, 123 syntactic –, 22 syntactic right –, 23 idempotent, 47, 123 image of a series, 39 inertia theorem, 109 integral domain, 99 integral part of a rational fraction, 53 invertible series, 5 irreducible set of matrices, 91 language, 2, 35 proper –, 37 rational –, 35 recognizable –, 35 leap-frog construction, 103 length – of a word, 2 average –, 127 letter, 2 linear recurrence relation, 32, 50 linear representation, 8 locally finite family, 4 matrix proper –, 13 Hankel –, 25 star of a –, 13 measure, 120 merge, 56 message, 115 minimal – automaton, 24 – denominator, 50

– polynomial, 50 module, 8 finitely generated –, 8 monoid – algebra, 47 free –, 2 morphism – of formal series, 16 – of semiring, 2 multiplicity of an eigenvalue, 50 normalized, 50 open peoblem, 70 open problem, 48, 88 p-adic valuation, 58 palindrome, 34, 40 periodic purely –, 63 quasi –, 63 poles, 50 polynomial, 3 – growth, 87, 94 exponential –, 54 minimal –, 50 support of an exponential –, 55 Post correspondence problem, 90, 96 prefix – -closed, 29 – code, 117 – set, 29 prime factors of a series, 58 prime subsemiring, 15 primitive polynomial, 111 product – of languages, 35 – of series, 3 Hadamard –, 11 proper – language, 37 – linear recurrence relation, 52 – matrix, 13 – series, 4 purely periodic, 63 quasi-integral, 72 quasi-periodic, 63 quasi-power, 43 quasi-regular, 17 quotient of a semiring, 15 rank

140 – of a series, 24 of a matrix, 25 rational – closure, 5 – language, 35 – operations, 5 – series, 5 R+ - – function, 75 unambiguous – operations, 70 rationally – closed, 5 – separated, 46 ray, 93 reciprocal polynomial, 51 recognizable – language, 35 – series, 7 reduced linear representation, 26 reduction algorithm, 33 regular – linear representation, 52 – rational series, 52 – semiring, 16 representation dimension of a linear –, 8 linear –, 8 reduced linear –, 26 regular – of a monoid, 36 tree –, 30 reversal, 34 right complete, 29 Schreier’s formula, 21 semiring, 1 – morphism, 2 Boolean –, 2 prime –, 15 regular –, 16 simplifiable –, 16 topological –, 4 semisimple, 132 separated rationally –, 46 series characteristic – of a language, 5 formal –, 2 morphism of formal –, 16 proper –, 4 rational –, 5 recognizable –, 7 shuffle product, 18 similar linear representations, 27 simple

Index – elements, 55 – set of recognizable series, 57 simplifiable semiring, 16 stable, 9 star – -height, 88 – of a matrix, 13 – of a series, 4 submodule, 8 subsemiring, 1 suffix – -closed, 32 – set, 33 suffix code, 117 summable family, 4 support – of a series, 2 – of an exponential polynomial, 55 synchronizing, 125 syntactic – algebra, 22, 34 – congruence, 47 – ideal, 22 – monoid, 47 – right ideal, 23 thin, 132 topological semiring, 4 torsion-free, 72 tree representation, 30 trivial relation, 113 trivially a polynomial, 109 ultrametric distance, 3 unambiguous rational operations, 70, 117 undecidable problem, 90 uniform Bernoulli morphism, 120 weak algorithm, 99 word, 2 empty –, 2