Polynomial derivations and their rings of constants

Transcription

1 U N I W E R S Y T E T M I K O L A J A K O P E R N I K A Rozprawy Andrzej Nowicki Polynomial derivations and their rings of constants T O R U Ń

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3 Polynomial derivations and their rings of constants i Contents Introduction 1 Part I. Preliminary concepts and properties of polynomial derivations 7 1 Definitions, notations and basic facts Derivations Derivations in polynomial rings Derivations in fields of rational functions Algebraic field extension and derivations Derivations in power series rings Systems of differential equations Useful facts and preliminary results Homogeneous derivations Darboux polynomials The divergence and special derivations Automorphism E d Bases of derivations in polynomial and power series rings The image of derivations Part II. Characterization of subalgebras of the form A d 39 3 Characterization of subfields Initial observations Derivations of purely transcendental field extensions Algebraically closed subfields Characterization of subalgebras Integrally closed subrings Rings of invariants Part III. Finiteness and properties of A d 50 5 General properties of the rings of constants for polynomial derivations Extension of scalars Closed polynomials First integrals and the ring of constants

4 ii Andrzej Nowicki 6 Constants for locally nilpotent derivations The function deg d and the automorphism e d The theorem of Weitzenböck The example of Deveney and Finston Principal elements Derivations without principal elements Results of van den Essen On van den Essen s algorithm Generating sets for some Weitzenböck derivations Weitzenböck derivations with 2 2 cells Comments and remarks Rings of constants for small n Finiteness for n = 1, 2 and Two variables Examples of derivations with trivial ring of constants Minimal generators Comments and remarks Part IV. Locally finite derivations 93 8 Farther properties of locally nilpotent derivations On the equality d(a) = ua The derivation ad + bδ The theorems of Rentschler Comments and remarks Local finiteness Locally finite endomorphisms Equivalent conditions Examples Jordan-Chevalley decomposition Semisimple derivations Polynomial flows The divergence of locally finite derivations Comments and remarks Part V. Polynomial derivations with trivial constants 116

5 Polynomial derivations and their rings of constants iii 10 Rational constants of linear derivations The main results Linear derivations Proof of Theorem Proof of Theorem A theorem of Jouanolou Degree and multiplicities of plane algebraic curves Darboux points Proof of Jouanolou s theorem: initial part Local analysis Global analysis Conclusion of the proof: first case The second case Comments and remarks Some applications of the local analysis Factorisable derivations A useful determinant An example of factorisable system Another example Simple polynomial derivations Properties of simple derivations Shamsuddin s result Derivation D(a, b) Examples of simple derivations References 158 Index 167

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7 Introduction The fundamental relations between the operation of differentiation and that of addition and multiplication of functions have been known for as long a time as the notion of the derivative itself. The relations were deepened when it was found that the operation of differentiation of functions on the smooth varieties with respect to a given tangent field not only has the formal properties of differentiation but also conversely; the tangent field is fully characterized by such an operation. Therefore, it was possible to define e. g. the tangent bundle in terms of sheaves of functions. The notion of the ring with derivation (=differentiation) is quite old and plays a significant role in the integration of analysis, algebraic geometry and algebra. In the 1940s it was found that the Galois theory of algebraic equations can be transferred to the theory of ordinary linear differential equations (the Picard - Vessiot theory). The field theory also included the derivations in its inventory of tools. The classical operation of differentiation of forms on varieties led to the notion of differentiation of singular chains on varieties, a fundamental notion of the topological and algebraic theory of homology. In 1950s a new part of algebra called differential algebra was initiated by the works of Ritt and Kolchin. In 1950 Ritt [93] and in 1973 Kolchin [48] wrote the well known books on differential algebra. Kaplansky, too, wrote an interesting book on this subject in 1957 ([42]). The present paper deals with k-derivations of the polynomial ring k[x] = k[x 1,..., x n ] over a field k of characteristic zero. The object of principal interest in this paper is k[x] d, the ring of constants of a k-derivation d of k[x], that is, k[x] d = {f k[x]; d(f) = 0}. Assume that f 1,..., f n are polynomials belonging to k[x]. There exists then (see [8]) a unique k-derivation d of k[x] such that d(x 1 ) = f 1,..., d(x n ) = f n. The derivation d is defined by d(h) = f 1 h x f n h x n, (0.1) for h k[x]. Consider now a system of polynomial ordinary differential equations dx i (t) dt = f i (x 1 (t),..., x n (t)), 1 i n. (0.2) If k is a subfield of the field C of complex numbers, then it is evident what the system means. When k is arbitrary then it also has a sense. This 1

8 2 Introduction system has a solution in k[[t]], the ring of formal power series over k in the variable t (see Section 1.6). Let k(x) = k(x 1,..., x n ) be the quotient field of k[x]. An element h of k[x] k (resp. of k(x) k) is said to be a polynomial (resp. rational) first integral of the system (0.2) if the following identity holds f 1 h x f n h x n = 0. (0.3) Thus, the set of all the polynomial first integrals of (0.2) coincides with the set k[x] d k where d is the k-derivation defined by (0.1). Moreover, the set of all the rational first integrals of (0.2) coincides with the set k(x) d k, where k(x) d = {h k(x); d(h) = 0} and where d is the unique extension of the k-derivation (0.1) to k(x). In various areas of applied mathematics (as well as in theoretical physics and chemistry) there occur autonomous systems of ordinary differential equations of the form (0.2). There arises the following question: do there exist first integrals of a certain type, for example, polynomial or rational first integrals?. This problem has been studied intensively for a long time; see for example [103], [97], [63] and [32] where many references on this subject can be found. The problem is known to be difficult even for n = 2. Computers are frequently used in solving this problem. There are computer programmes which make it possible to find all the polynomial first integrals up to a given highest degree r but they do not provide any information beyond r. Throughout the paper we use the vocabulary of differential algebra ([42], [48]). In terms of derivations the above problem consists in the finding of methods leading to the statement whether the ring of the form k[x] d (or k(x) d ), where d is a given k-derivation of k[x], is nontrivial, i. e., different than k. A certain result containing some necessary and sufficient conditions (even for n = 2) on polynomials f 1,..., f n would be desirable and remarkable for the derivation defined by the formula (0.1) to possess a nontrivial ring of constants. There exist other natural problems concerning the discussed question. Assume that d is a k-derivation of k[x] such that k[x] d k. Then there arises the following question: Is the ring k[x] d finitely generated over k? This question is a special case of the fourteenth problem of Hilbert ([66], [35]). Let us stress that there exist k-derivations of k[x] for which the

9 Introduction 3 ring of constants is not finitely generated (see Section 4.2). How to decide whether a given k-derivation of k[x] has a finitely generated ring of constants? Suppose that we already have one such derivation which has a finitely generated ring of constants. How can one find its finite (possibly smallest) generating set? Can the minimal number of generators be limited in advance? What can be said about this number? Evidently, not every k-subalgebra of k[x] is a ring of constants with respect to a certain k-derivation (or a family of k-derivations) of k[x]. For example, k[x 2 1,..., x2 n] is such a subalgebra. Therefore a question arises which subalgebras are the rings of constants. Does there exist an algebraic description of such subalgebras? Let D be a family of k-derivations of k[x]. Consider the ring of constants k[x] D = k[x] d = {w k[x]; d(w) = 0 for all d D}. d D Does there exist a k-derivation δ of k[x] such that k[x] D = k[x] δ? Similar questions can be asked for all the subfields of the field k(x). All the above questions will constitute a group of main problems dealt with in the present paper. We will also be preoccupied with other issues related to the constant rings in k[x]. The paper contains the author s results concerning derivations (not only in polynomial rings) closely connected with the rings of constants. In particular, we present: a) methods leadings to the proof that some polynomial derivations do not possess a nontrivial polynomial (often even rational) constant as well as methods for the finding of a finite set of generators, illustrated by numerous examples; b) an algebraic description of all the subrings of k[x] which are rings of constants of derivations. Moreover, applications of the description to the above mentioned problems of the finiteness and the minimal number of generators. This thesis is divided into 13 chapters grouped in 5 parts. Let us briefly present the main author s results contained in this paper. One of the main results is Theorem describing all the subrings of a finitely generated k-algebra A (without zero divisors) which are rings of

10 4 Introduction constants with respect to derivations. We show that a k-subalgebra B of A is of the form A d if and only if B is integrally closed in A and B 0 A = B, where B 0 is the field of fractions of B. By this theorem it is easy to prove (Theorem 4.1.5) that if D is a family of k-derivations of A, then A D = A d, for some k-derivation d of A. As a consequence of the theorem we get Theorem which states that if G GL n (k) is a connected algebraic group, then there exists a k-derivation d of k[x] such that the invariant ring k[x] G is equal to k[x] d. Using this fact and some known facts related to the fourteenth Problem of Hilbert one can easily deduce that if n 7, then there always exists a k-derivation of k[x] such that the ring of constants k[x] d is not finitely generated over k (see Section 4.2). The question of what happens for n < 7 is an open one. In Section 7.1 we show how it follows from a result of Zariski [113] that if n 3 then every ring of the form k[x] d is finitely generated. For the first time it was observed by Nagata and the author in [85] in We show (in Section 7.1) that if n = 2 then k[x] d is of the form k[f], for some f k[x]. This means that every ring of constants in the polynomial ring in two variables is generated by one polynomial. In Section 7.4 we prove that if n 3 then the minimal number of generators is unbounded. Moreover, we show (see Section 5.2) that any minimal generating set of k[x] d has a special property. Every element of such a set is the so called closed polynomial. Properties and applications of closed polynomials are described in Section 7.2 devoted to the derivations of k[x, y]. In this paper much attention is paid to the k-derivations of k[x] such that k[x] d = k or k(x) d = k. All the linear k-derivations having this property are characterized in Chapter 10. Inspired by the proof of Jouanolou s nonintegrability theorem [40], we describe in Chapter 11 a method proving the nonexistence of nontrivial constants for some k-derivations of k[x]. Several examples, including Jouanolou s one, are described in details. Let us note that the original proof of Jouanolou s theorem was incomplete (this will be explained in the introduction to Chapter 11). The whole Chapter 12, where we concentrate on the so called factorisable derivations, is also devoted to this method. Another method is presented in the proofs of Examples and The results concerning this aim can also be found in Chapter 13 where we present an algorithm for a verification whether a given derivation is simple. Using this algorithm we obtain new examples of k-derivations in k[x] without rational constants.

11 Introduction 5 Here are some other important results of the author, concerning polynomial derivations and contained in this paper. In Section 2.6 we prove that if L is a field containing k with tr.deg k L < and d is a k-derivation of L, then the image of d is different than L. This fact is useful to a construction of k-derivations of k[x] with the trivial ring of constants (see Section 3.2). The existence of such derivations leads to Theorem which states that if k L is a purely transcendental field extension, then there exists a derivation d of L such that L d = k. As a consequence we get (Theorem 3.3.2): If k L is an arbitrary field extension, then an intermediate field M is of the form L d if and only if M is algebraically closed in L. This theorem is an extension of results of Suzuki [107] and Derksen [16]. If d is a k-derivation of k[x], then we denote by d the divergence of d, that is, d = n i=1 d(x i)/ x i. A derivation d of k[x] is called special if d = 0. Some initial properties of the divergence and special derivations are given in Sections 2.3. Every locally nilpotent k-derivation of k[x] is special (see Theorem 9.7.5). In Section 2.5 we describe all the bases and all the commutative bases of the free k[x]-module Der k (k[x]). We prove that every component of a commutative basis of Der k (k[x]) is a special derivation. In Section 7.2 (see Theorem ) we prove that if d is a k-derivation of k[x, y], then k[x, y] d k if and only if the derivation d is similar to a special k-derivation δ (that is, there exists nonzero elements a, b k[x, y] such that ad = bδ). Consider a derivation (of an algebra A) of the form = ad + bδ, where a, b A and where d, δ are locally nilpotent derivations of A which commute. Section 8.2 is devoted to the question of finding necessary and sufficient conditions on a and b for to be locally nilpotent. Theorem 8.2.1, which is the main result of Section 8.2, gives a partial answer to this question. In Chapter 9 (see Corollary 9.4.7) we show that if d is a linear homogeneous k-derivation of k[x], then the algebra Nil(d) = {w k[x]; d s (w) = 0, for some s N} is finitely generated over k. The paper also contains new (shorter and easier) proofs of some known theorems and facts concerning derivations in polynomial rings and their rings of constants. We show in Chapters 6 and 8 that the proofs of many known theorems on locally nilpotent derivations can be simplified by the introduction of

12 6 Introduction function deg d (Section 6.1) and by using its properties. A short proof of Theorem and proofs of the facts which are consequences of this theorem (see for example Lequain s Theorem 8.1.6) are worth mentioning. In Section 7.3 (Example 7.3.1, Example 7.3.2) we present simple proofs of theorems of Schwarz [97] concerning the Selkov and Guerilla Combat systems. Section 9.6 contains a very short proof of Coomes and Zurkowski s Theorem which states that a flow is polynomial if and only if the derivation associated with it is locally finite. In Section 13.2 we present a proof of Shamsuddin s Theorem It is difficult to find an original proof of this important fact. We know the theorem from [38], where it is only mentioned without proof. In Section 6.7 we recall van den Essen s algorithm based on the theory of Gröbner bases. With the help of this algorithm, using the CoCoA programme [3], we give (in Sections 6.8 and 6.9) a series of examples related to the finite generating sets of the Weitzenböck derivations. Let us note that in some cases such generators have not been found yet. The present paper is mainly based on the author s papers: [79], [81], [83], [84], and on papers: [28], [47], [64], [85], [87] written jointly with other authors.

13 Part I Preliminary concepts and properties of polynomial derivations 1 Definitions, notations and basic facts Throughout the paper all rings are commutative with identity. N and N 0 denote the set {1, 2,... } of natural numbers and the set {0, 1, 2,... } of nonnegative integers, respectively. A ring R is called Z-torsion free if the equality na = 0, where n N and a R, implies that a = 0. A ring R is called reduced if R has no nonzero nilpotent elements. If a k-algebra A (where k is a ring) has no zero divisors, then we say that A is a k-domain and we denote by A 0 its field of fractions. If (i 1,..., i s ) is a sequence of nonnegative integers, then we denote by i 1,..., i s the Newton number (i i s )!(i 1! i s!) 1. This chapter is an introductory one. Section 1.1 contains basic definitions concerning derivations, differential algebra and rings of constants. In the next sections we present in a concise manner basic facts on derivations in polynomial rings, fields of rational functions and rings of power series. Section 1.6 is devoted to the formal solutions of systems of ordinary differential equations. For the proofs of the facts which are not proved here, the reader is asked to refer to [115], [69], [52] and [8]. In this chapter (unless otherwise stated) k and R are always rings. 1.1 Derivations An additive mapping d : R R is said to be a derivation of R if, for all x, y R, d(xy) = xd(y) + d(x)y. We denote by Der(R) the set of all derivations of R. If d, d 1, d 2 Der(R) and x R, then the mappings xd, d 1 + d 2 and [d 1, d 2 ] = d 1 d 2 d 2 d 1 are also derivations. Thus, the set Der(R) is an R-module which is also a Lie algebra. Note the following simple propositions 7

14 8 Part I. Preliminary concepts Proposition If d is a derivation of R, x 1,..., x s R and n 0, then d n (x 1 x s ) = i 1,..., i s d i 1 (x 1 ) d is (x s ). i 1 + +i s=n Proposition Let d 1, d 2 be derivations of R and let A = {x R; d 1 (x) = d 2 (x)}. Then A is a subring of R. If R is a field, then A is a subfield of R. If R is a k-algebra, then a derivation d of R is called a k-derivation if d(αx) = αd(x) for α k and x R. We denote by Der k (R) the set of all k-derivations of R. A differential k-algebra is a pair (R, d), where R is a k-algebra and d is a k-derivation of R. Let (R 1, d 1 ) and (R 2, d 2 ) be differential k-algebras and let f : R 1 R 2 be a homomorphism of k-algebras. Put T (f) = {x R 1 ; fd 1 (x) = d 2 f(x)}. The homomorphism f is called differential if fd 1 = d 2 f, i. e., if T (f) = R 1. The set T (f) is a k-subalgebra of R 1. As a consequence of this fact we get Proposition If R 1 = k[a 1,..., a s ] is finitely generated over k, then f is differential iff a 1,..., a s T (f). Assume now that (R, d) is a differential k-algebra. An ideal A of R is said to be a differential ideal or a d-ideal if d(a) A. Let A be a differential ideal of R and consider the quotient k-algebra R = R/A. There exists a unique k-derivation d of R such that the natural homomorphism R R is differential. The derivation d is defined by d(x + A) = d(x) + A, for all x R. Let S be a multiplicatively closed subset of R and let R S be the k- algebra of fractions of R with respect to S. Then there exists a unique k-derivation d S of R S such that the natural homomorphism R R S, r r/1, is differential. The derivation d S is defined by the formula: d S (r/s) = (d(r)s rd(s))/s 2, for r R and s S. Let D be a family of derivations of R. We denote by R D the set {x R; d(x) = 0 for any d D}. This set is a subring of R. We call it the ring of constants of R (with respect to D). If R is a k-algebra and D is a family of k-derivations of R, then R D is a k-subalgebra of R. If R is a field, then R D is a subfield of R. If D has only one element d, then we write R d instead of R {d}. It is clear that R D = d D Rd.

15 Chapter 1. Definitions, notations and basic facts Derivations in polynomial rings Let R[X] = R[x i ; i I] be a polynomial ring over a k-algebra R. For each i I the partial derivative x i is an R-derivation of R[X]. It is (by Proposition 1.1.2) a unique R-derivation d of R[X] such that d(x i ) = 1 and d(x j ) = 0 for all j i. Assume that d is a k-derivation of R. Then there exists a unique k- derivation d of R[X] such that d R = d and d(x i ) = 0 for all i I. Moreover, if f : I R[X] is a function, then there exists a unique k- derivation D of R[X] such that D R = d and D(x i ) = f(i) for all i I. The derivation D is defined as follows: if w R[X], then D(w) = d(w) + f(i 1 ) w x i1 + f(i n ) w x in, where {i 1,..., i n } is a finite subset of I such that w R[x i1,..., x in ]. As a consequence of the above facts we get Theorem Let k[x] = k[x 1,..., x n ] be a polynomial ring over a ring k. (1) If f 1,..., f n k[x], then there exists a unique k-derivation d of k[x] such that d(x 1 ) = f 1,..., d(x n ) = f n. This derivation is of the form: d = f 1 x f n x n. (2) Der k (k[x]) is a free k[x]-module on the basis (3) x i x j = x j x i, for all i, j {1,..., n}. (4) If d Der k (k[x]) and f k[x], then d(f) = n i=1 Note also the following x 1,..., x n. f x i d(x i ). Proposition Let d be a k-derivation of a k-algebra A. Let f k[x] = k[x 1,..., x n ] and a = (a 1,..., a n ) A n. Then f(a) is an element of A and d(f(a)) = n f i=1 x i d(a i ). Proof. Put M = {g k[x]; d(g(a)) = g x 1 d(a 1 ) + + g x n d(a n )}. It is easy to check that M is a k-subalgebra of k[x] containing x 1,..., x n. Thus, M = k[x] and hence, f M. Assume now that R is a Z-torsion free ring. Let R[t] be the polynomial ring over R in one variable t, and let d = t. Then it is easy to see that R[t] d, the ring of constants of R[t] with respect to d, is equal to R. In particular, we have the following

16 10 Part I. Preliminary concepts Proposition If k is Z-torsion free and k[t, x 1,..., x n ] is the polynomial ring, then k[t, x 1,..., x n ] d = k[x 1,..., x n ], where d = t. By the above proposition and Proposition 1.1.3, we get Proposition Assume that k is Z-torsion free. Let A = k[t, x 1,..., x n ], B = k[u, y 1,..., y m ] be the polynomial rings over k. Put d A = t, d B = u and let f : A B be a homomorphism of k-algebras. Then the following conditions are equivalent: (1) f is differential (that is, fd A = d B f). (2) The polynomials f(t) u, f(x 1 ),..., f(x n ) belong to k[y 1,..., y m ]. Proof. (1) (2). In view of Proposition we know, that k[y 1,..., y m ] = B d B. So, we must show that d B (f(t) u) = d B (f(x 1 )) = = d B (f(x n )) = 0. If i {1,..., n}, then d B (f(x i )) = fd A (x i ) = f(0) = 0. Moreover, d B (f(t) u) = fd A (t) d B (u) = f(1) 1 = 0. (2) (1). The variables t, x 1,..., x n belong to T (f) = {a A; fd A (a) = d B f(a)} and hence (by Proposition 1.1.3), f is differential. 1.3 Derivations in fields of rational functions Using a simple modification of the proofs of facts presented in the previous section, one can prove analogous properties for the fields of rational functions. In particular, we have the following three propositions. Proposition Let X be an algebraically independent set over a field R and let R(X) be the purely transcendental field extension of R. If d is a derivation of R and f : X R(X) is a function, then there exists a unique derivation D of R(X) such that D R = d and D(x) = f(x) for all x X. Proposition Let k(x) = k(x 1,..., x n ). (1) If f 1,..., f n k(x), then there exists a unique k-derivation d of k(x) such that d(x 1 ) = f 1,..., d(x n ) = f n. This derivation is of the form: d = f 1 x f n x n. (2) The derivations x 1,..., form a basis of the k(x)-space Der k (k(x)). (3) x i x j = x j x i x n for all i, j {1,..., n}. (4) If d Der k (k(x)) and f k(x), then d(f) = n i=1 f x i d(x i ).

17 Chapter 1. Definitions, notations and basic facts 11 Proposition Let k L be fields. Let d be a k-derivation of L and let a = (a 1,..., a n ) L n. If F = f/g is an element of k(x 1,..., x n ) such that g(a) 0, then F (a) is an element of L and d(f (a)) = F x 1 d(a 1 ) + + F x n d(a n ). Assume now that L is a field of characteristic zero and consider L(t), the field of rational functions over L in one variable t. Let d = t and let f = a/b (where a and b are coprime polynomials in L[t]) be an element of L(t) such that d(f) = 0. Then ad(b) = d(a)b and hence, a d(a) and b d(b). Since deg d(a) < deg a and deg d(b) < deg b, d(a) = d(b) = 0 and hence (see Section 1.2), a, b L, i. e., f L. Thus, L(t) d = L. As a consequence of this fact we get Proposition Suppose that k is a field of characteristic zero and let d = t. Then k(t, x 1,..., x n ) d = k(x 1,..., x n ) and k(t)[x 1,..., x n ] d = k[x 1,..., x n ]. 1.4 Algebraic field extension and derivations We recall here some well known facts concerning derivations and algebraic field extensions of characteristic zero. Proofs of these facts can be found in [115], [69] or [52]. Theorem Let k L be fields of characteristic zero. The following conditions are equivalent: (1) L is algebraic over k. (2) For every derivation d of k there exists a unique derivation D of L such that D k = d. (3) If δ is a k-derivation of L, then δ = 0. Theorem Let L = k(p 1,..., p n ) be a finitely generated extension of a field k of characteristic zero. Put p = (p 1,..., p n ) and let M p = {f k[x 1,..., x n ]; f(p) = 0}. The following conditions are equivalent: (1) L is algebraic over k. (2) There exist n polynomials f 1,... f n M p such that det [ f i x j (p)] 0. Proposition Let L = k(x 1,..., x n ) be the field of rational functions over a field k of characteristic zero. If f 1,..., f n L, then the following

18 12 Part I. Preliminary concepts conditions are equivalent: (1) The elements f 1,..., f n are algebraically independent over k. (2) If d is a k(f 1,..., f n )-derivation of L, then d = 0. (3) det [ f i x j ] Derivations in power series rings Let k[[t ]] = k[[t 1,..., t n ]] be the power series ring over a ring k in n variables and let M denote the ideal (t 1,..., t n ). Let us recall that k[[t ]] is a complete ring with respect to the M-adic topology. Since m=0 M m = 0, the M-adic topology is Hausdorff. If d is a derivation of k[[t ]], then it is easy to check that d(m m ) M m 1, for every m N. This fact implies that every derivation of k[[t ]] is a continuous mapping and moreover, if d is a k-derivation of k[[t ]] such that d(t 1 ) = = d(t n ) = 0, then d = 0. Using the above facts and the same arguments as in Section 1.2, we get Proposition Let d be a derivation of k and let f 1,..., f n k[[t ]]. Then there exists a unique derivation D of k[[t ]] such that D k = d and D(t 1 ) = f 1,..., D(t n ) = f n. The derivation D is defined by the formula: D = d + f 1 t f n t n. Theorem ([8]). (1) If f 1,..., f n k[[t ]], then there exists a unique k-derivation d of k[[t ]] such that d(t 1 ) = f 1,..., d(t n ) = f n. The derivation d is of the form: d = f 1 t f n t n. (2) Der k (k[[t ]]) is a free k[[t ]]-module on the basis t 1,..., t n. (3) t i t j = t j t i, for all i, j {1,..., n}. (4) If d is a k-derivation of k[[t ]], then d(f) = f t 1 d(t 1 )+ + f t n d(t n ), for every f k[[t ]]. Now we will explain two differential formulas concerning rings of power series. Denote by Ω = Ω n the set {α = (α 1,..., α n ); α 1,..., α n N 0 }. If α = (α 1,..., α n ) is an element of Ω, then we denote by T α the monomial t α 1 1 tαn n. In particular, T 0 = t 0 1 t0 n = 1 and T α T β = T α+β for any α, β Ω. Every element of k[[t ]] has a unique decomposition of the form α a αt α, where a α k for all α Ω. If α = (α 1,..., α n ) Ω, then α denotes the sum α 1 + +α n. The ideal M m is generated by the set {T α ; α = m}. If m N and a α T α M m, then a α = 0, for all α such that α < m.

19 Chapter 1. Definitions, notations and basic facts 13 Put R = k[[t ]] and consider a second power series ring R[[Y ]] = R[[y 1,..., y s ]] = k[[t 1,..., t n, y 1,..., y s ]]. Let f = β Ω s f β Y β = β Ω s f β y β 1 1 yβs s be a series in R[[Y ]] and let ϕ 1,..., ϕ s be series, without constant terms, belonging to R. If p N, then put C p = β <p f βϕ β = β <p f βϕ β 1 1 ϕβs s. It is clear that C p R. Since ϕ 1,..., ϕ s have no constant terms, C p+1 C p M p. This implies that (C p ) is a Cauchy sequence in R and hence, this sequence is convergent. We denote by β Ω s f β ϕ β or by f(ϕ) = f(ϕ 1,..., ϕ s ) the limit of (C p ). Now we may state the following power series analogy of Proposition Proposition Let d be a derivation of R = k[[t ]], let f R[[Y ]], and let ϕ 1,..., ϕ s be series in R without constant terms. Put ϕ = (ϕ 1,..., ϕ s ). Then f(ϕ) is an element of R and d(f(ϕ)) = d(f)(ϕ) + s f i=1 y i (ϕ)d(ϕ i ), (1.1) where d is the derivation of R[[Y ]] defined by d( β f βy β ) = β d(f β)y β. f Proof. We already know that f(ϕ), y 1 (ϕ),..., f y 1 (ϕ) are well defined elements of R. We must prove only the equality (1.1). It suffices to prove that the element A = d(f(ϕ)) d(f)(ϕ) s f i=1 y i (ϕ)d(ϕ i ) (1.2) belongs to M p, for any p N. Let p N and let f = β f βy β. Set E p = f β Y β, β >p F p = f β Y β, β p E p (ϕ) = f β ϕ β, β >p F p (ϕ) = f β ϕ β. β p Then f = E p + F p, f(ϕ) = E p (ϕ) + F p (ϕ), and E p (ϕ) M p+1 and hence, d(e p (ϕ)) M p. Consequently, the elements d(e p )(ϕ) and Ep y i (ϕ) belong to M p, for every i = 1,..., s. Since F p is a polynomial in R[y 1,..., y s ], we have (see Section 1.2): d(f p (ϕ)) d(f p )(ϕ) s F p i=1 y i (ϕ)d(ϕ i ) = 0.

20 14 Part I. Preliminary concepts Therefore, the element A, defined by (1.2), is equal to d(e p (ϕ)) d(e p )(ϕ) s E p i=1 y i (ϕ)d(ϕ i ), and so, it is an element of M p. Corollary Let d be a k-derivation of k[[t ]], let f k[[y 1,..., y s ]] and let ϕ = (ϕ 1,..., ϕ s ) k[[t ]] s. Assume that the series ϕ 1,..., ϕ s have no constant terms. Then f(ϕ) k[[t ]] and d(f(ϕ)) = s i=1 f y i (ϕ)d(ϕ i ). Assume now that {B α } α Ω is a family of elements of k[[t ]]. Then there exists an element of k[[t ]] of the form α Ω B αt α. This element is the limit of the Cauchy sequence (C p ), where C p = α p B αt α. If d is a derivation of k[[t ]], then {d(b α )} α Ω is a family of elements in k[[t ]] and hence, we have the element of the form α Ω d(b α)t α. Moreover, we have also the element α B αd(t α ). Proposition d( α B αt α ) = α d(b α)t α + α B αd(t α ). Proof. Use the same argument as in the proof of Proposition Systems of differential equations In this section we prove two theorems concerning formal solutions of systems of ordinary differential equations which will be useful in the next chapters of this paper. Throughout this section k is a ring containing Q, k[x] = k[x 1,..., x n ] is the polynomial ring and k[[t]], k[x][[t]] are power series rings over k and k[x], respectively. The following theorem is a power series version of the well known Cauchy and Picard theorem from the theory of ordinary differential equations (see for example [90] or [106]). Theorem Let f 1,..., f n k[x][[t]] and a 1,..., a n k. Then there exist unique series ϕ 1,..., ϕ n k[[t]] such that: (1) ϕ i t = f i (ϕ 1,..., ϕ n ), for i = 1,..., n, and (2) the constant terms of ϕ 1,..., ϕ n are equal to a 1,..., a n, respectively.

21 Chapter 1. Definitions, notations and basic facts 15 For the proof of this theorem we need some notations and lemmas. Assume that B 0, B 1,... is a sequence of series belonging to k[[t]] and put C p = p i=1 B it i for any p N 0. Since C p+1 C p (t) p, these series form a Cauchy sequence in k[[t]] and hence, there exists the series p=0 B pt p. It is easy to check the following Lemma If B p = i=0 b pit i with b pj k, then p=0 B pt p = s=0 a st s, where a s = i+j=s b ij for any s N 0. Let f be a series in k[x][[t]] and let ϕ 1,..., ϕ n k[[t]]. Set ϕ = (ϕ 1,..., ϕ n ) and let f = p=0 f pt p, where each f p is a polynomial in k[x]. Then we have the sequence f 0 (ϕ), f 1 (ϕ),... of series in k[[t]] and so, we can form the series f(ϕ) = p=0 f p(ϕ)t p. When the ring k has no zero divisors then the following lemma is easy to be proved. We present a proof of it in the general case. Lemma If g is a nonzero polynomial in k[x], then there exists a point a k n such that g(a) 0. Proof. Let us recall that k[x] = k[x 1,..., x n ] and Q k. First assume that n = 1. Set x = x 1 and let g = p j=0 g jx j with g j k and g p 0. If p = 0 then our lemma is evident. Let p > 0 and suppose that g(a) = 0 for all a k 1. Put h(x) = 2 p g(x) g(2x). Then h(a) = 0 (for all a k 1 ) and deg h < deg g. So, by induction, h = 0. This implies that g 0 = = g p 1 = 0 and we have a contradiction: 0 = g(1) = g p 0. This completes the proof for n = 1. Assume now that n > 1 and let g = p j=0 g jx j n with g j k[x 1,..., x n 1 ] and g p 0. Then, by induction, there exists b k n 1 such that g p (b) 0. Consider the polynomial g(b, x n ). This is a nonzero polynomial in a one variable. Therefore, by the first part of this proof, g(b, a n ) 0, for some a n k. As a consequence of the above lemma we get Lemma Let f, g k[x][[t]]. If f(a) = g(a) for any a k n, then f = g. If ψ = i=0 a it i k[[t]] and p N 0, then denote by ψ [p] the p-th coefficient of ψ (that is, ψ [p] = a p ). Moreover, denote by s p (ψ) the sum p i=0 a it i.

22 16 Part I. Preliminary concepts Lemma Let f k[x][[t]], p N 0 and ϕ 1,..., ϕ n k[[t]]. Then f(ϕ 1,..., ϕ n ) [p] = f(s p (ϕ 1 ),..., s p (ϕ n )) [p]. Proof. Observe that if ϕ and ψ are two series in k[[t]], then (ϕ+ψ) [p] = (s p (ϕ) + s p (ψ)) [p] and (ϕ ψ) [p] = (s p (ϕ) c p (ψ)) [p]. So Lemma holds for f k[x]. The rest follows from Lemma Proof of Theorem We construct the coefficients (ϕ 1 ) [p],..., (ϕ n ) [p] using induction on p. If p = 0 then put (ϕ 1 ) [0] = a 1,..., (ϕ n ) [0] = a n. (1.3) Let p 0 and assume that, for all j p, the coefficients (ϕ 1 ) [j],..., (ϕ n ) [j] are already constructed. If i {1,..., n}, then we define: (ϕ i ) [p+1] = 1 p + 1 f i(g 1,..., g n ) [p], where g i = p (ϕ i ) [j] t j. (1.4) Thus, we have constructed the series ϕ 1,..., ϕ n k[[t]] (where ϕ i = p=0 (ϕ i) [p] t p for i = 1,..., n) satisfying (2). It is easy to check (using Lemma 1.6.5) that they also satisfy (1). Assume now that ϕ = (ϕ 1,..., ϕ n ) is an arbitrary sequence of series in k[[t]] satisfying (1) and (2). Then ( ϕ i t ) [p] = f i (ϕ 1,..., ϕ n ) [p], for any p N 0 and i = 1,..., n. Hence (by Lemma 1.6.5), ϕ satisfies the equalities (1.4). Since ϕ also satisfies (1.3), we see (by a simple induction) that ϕ is unique. If f = (f 1,..., f n ) is a sequence of series in k[x][[t]] and a = (a 1,..., a n ) k n, then denote by ϕ(t, a) = (ϕ 1 (t, a),..., ϕ n (t, a)) the unique sequence (ϕ 1,..., ϕ n ) of series from k[[t]] satisfying the conditions (1) and (2) of Theorem The sequence ϕ(t, a) is called the formal solution of the differential system j=0 X t = f(x), X [0] = a. (1.5)

23 Chapter 1. Definitions, notations and basic facts 17 Now we prove the following polynomial property of the formal solutions. Theorem If f k[x][[t]] n, then there exist uniquely determined polynomials ω ij k[x] (for i {1,..., n} and j N 0 ) such that ϕ i (t, a) = ω ij (a)t n, (1.6) j=0 for all a k n and i {1,..., n}, where ϕ(t, a) is the formal solution of (1.5). Proof. Similarly as in the proof of Theorem we construct (using an induction on p N 0 ) the polynomials ω 1p, ω 2p,..., ω np. If p = 0 then put ω 01 = x 1,..., ω 0n = x n. Let p 0 and assume that, for all j p, the polynomials ω 1j,..., ω nj are already defined. Then define: ω i(p+1) = 1 p + 1 f i(g 1,..., G n ) [p], where G i = p j=0 ω ijt j for any i = 1,..., n. Now, using Lemma 1.6.5, one can easily deduce that the polynomials ω ij satisfy (1.6). The uniqueness follows from Lemma and Theorem If a = (a 1,..., a n ) k n, then we denote by π a the surjective homomorphism from k[x][[t]] to k[[t]] defined by π a (f) = f(a). Corollary If f k[x][[t]] n, then there exist unique series W 1,..., W n k[x][[t]] such that ϕ i (t, a) = π a (W i ) for all a k n and i {1,..., n}.

24 2 Useful facts and preliminary results This chapter is devoted to the general properties of derivations in polynomial rings. We present some concepts and facts which will be often used in the next chapters. We have here six sections concerning different, to some extent independent, matters. We assume that n is a fixed natural number and we use the abbreviated denotations: k[x] = k[x 1,..., x n ] and k(x) = k(x 1,..., x n ) for the ring of polynomials and the field of rational functions, respectively. By a direction we mean a nonzero sequence γ = (γ 1,..., γ n ) of integers. In Section 2.1 we first characterize all the polynomials and rational functions which are homogeneous with respect to a direction, and then, we give some information on homogeneous derivations and, in particular, on monomial derivations, that is, on derivations d of k[x] such that d(x 1 ),..., d(x n ) are monomials. Section 2.2 contains the basic facts concerning Darboux polynomials for k-derivations of k[x]. We prove here, among other things, a useful proposition (see Proposition 2.2.4) which states that every homogeneous k-derivation of k[x, y] has a Darboux polynomial. If d is a k-derivation of k[x], then we denote by d the divergence of d, that is, d = n i=1 d(x i)/ x i. A derivation d of k[x] is called special if d = 0. In Section 2.3 we present some initial properties of the divergence and special derivations. More such properties and their applications will be given in the next sections. Section 2.4 deals with the automorphism E d = exp(td) of the power series ring k[x][[t]]. First we recall some well known facts concerning this automorphism and its applications to the autonomous systems of differential equations, and then, we study the jacobian of E d. In Section 2.5 we concentrate on the bases of the free R-module Der k (R), where R = k[x] or k[[x]]. We present descriptions of all the bases and all the commutative bases of Der k (R). The descriptions come from the author s paper [79]. Moreover, we prove (Theorem 2.5.5) that every component of a commutative basis of Der k (k[x]) is a special derivation. The whole of Section 2.6 is a copy of paper [47], by K. Kishimoto and the author, devoted to the images of derivations in k(x). Assuming that k is a field of characteristic zero, we prove that if d is a k-derivation of k(x) (in particular, of k[x]), then d((x)) k(x). We obtain it as a consequence 18

25 Chapter 2. Useful facts and preliminary results 19 of a more general theorem which states that if d is a k-derivation of a field L containing k with tr.deg k L <, then d(l) L. Note that our theorem is important for the considerations contained in Chapter 3. Using this theorem it is easy to construct k-derivations d of k[x] such that k(x) d = k. 2.1 Homogeneous derivations Let γ = (γ 1,..., γ n ) be a direction and let s Z. If α = (α 1,..., α n ) Ω = N n 0, then γα denotes the sum γ 1α γ n α n and X α denotes the monomial x α 1 1 xαn n. A nonzero polynomial f k[x] is said to be a γ-form of degree s (or a γ- homogeneous polynomial of degree s) if f is of the form: f = γα=s a αx α, where a α k. We assume that the zero polynomial is a γ-form of any degree. For example, if n = 2 and k[x] = k[x, y], then y 5 + xy 3 6x 2 y is a (2, 1)-form of degree 5, x 4 y + x 7 y 3 is a ( 2, 3)-form of degree 5 and x 8 y 4 + x 6 y is a (1, 2)-form of degree 0. Proposition Assume that k is a domain of characteristic zero and k 0 is its field of fractions, If f is a nonzero polynomial in k[x], then the following conditions are equivalent: (1) f is a γ-form of degree s. (2) f(t γ 1 x 1,..., t γn x n ) = t s f(x 1,..., x n ) (in the ring k 0 (t)[x]). f f (3) γ 1 x 1 x γ n x n x n = sf. Proof. (3) (2). Set R = k 0 (t)[x] = k 0 (t)[x 1,..., x n ], u = (t γ 1 x 1,..., t γn x n ), g = t s f(u), δ = t, and let ϕ : k 0[X] R be the homomorphism of k 0 -algebras defined by ϕ(x i ) = t γ i x i, for i = 1,..., n. Applying ϕ for (3) we get γ 1 t γ 1 x 1 f x 1 (u) + + γ n t γn x n f x n (u) sf(u) = 0. Next, using this equality and a simple calculation, we deduce that g R δ. This implies (by Proposition 1.3.4) that g is a polynomial in k 0 [X]. Thus, we have the equality: f(t γ 1 x 1,..., t γn x n ) = f(u) = t s g(x 1,..., x n ). Substituting now t = 1, we get: g(x 1,..., x n ) = f(x 1,..., x n ) and hence, we have (2). The implications (2) (1) and (1) (3) are well known and easy to be proved.

26 20 Part I. Preliminary concepts The implication (1) (3) (for γ = (1,..., 1) and k = R) is the well known Euler theorem on homogeneous functions ([24], [25]). Equality (3) is called the Euler formula. Denote by A (s) γ the group of all γ-forms of degree s. Each A (s) γ k-submodule of k[x] and k[x] = s Z A(s) γ. Moreover, A (s) γ A (t) γ is a A (s+t) γ for all s, t Z. Thus, k[x] is a graded ring. Such a gradation on k[x] is said to be a γ-gradation. Every polynomial f k[x] has the γ-decomposition f = f s into γ-components f s of degree s. If f 0, then γ-deg(f) denotes the γ-degree of f, that is, the maximal s such that f s 0. We assume also that γ-deg(0) =. It is easy to check the following Lemma Let k be a domain and let f, g be nonzero polynomials in k[x]. If fg is a γ-form, then f and g are γ-forms too. Assume now that k is a field (of characteristic zero). An element ϕ of k(x) is said to be γ-homogeneous of degree s if, in the field k(t, x 1,..., x n ), the following equality holds: ϕ(t γ 1 x 1,..., t γn x n ) = t s ϕ(x 1,..., x n ). Proposition Let f, g be nonzero coprime polynomials in k[x] and let ϕ = f/g. The following conditions are equivalent: (1) ϕ is γ-homogeneous of degree s. (2) Polynomials f and g are γ-forms of degrees p and q, respectively, where s = p q. ϕ ϕ (3) γ 1 x 1 x γ n x n x n = sϕ. Proof. (1) (2). Consider the γ-decompositions f = f p1 + +f p, g = g q1 + + g q, where p 1 < < p, q 1 < < q, and put X = (x 1,..., x n ), t γ X = (t γ 1 x 1,..., t γn x n ). By the equality ϕ(t γ X) = t s ϕ(x) we obtain the following equality of polynomials in k(t)[x]: Hence, by Proposition 2.1.1, f(t γ X)g(X) = t s f(x)g(t γ X). (t p 1 f p1 (X) + + t p f p (X))g(X) = t s f(x)(t q 1 g q1 (X) + + t q g q (X)).

27 Chapter 2. Useful facts and preliminary results 21 Comparing now powers of t we see that p = s + q and f p (X)g(X) = g q (X)f(X). Since the polynomials f and g are coprime, f p = fh and g q = gh for some h k[x] and hence, by Lemma 2.1.2, f and g are γ-forms. (2) (3) It is a consequence of Proposition 2.1.1(3). (3) (2) Use the same arguments as in the proof of the implication (3) (2) of Proposition The implication (2) (1) is obvious. A k-derivation d of k[x] is called γ-homogeneous of degree s if d(a (p) γ ) A (s+p) γ, for any p Z. For example, if k[x] = k[x, y] and d(x) = x + y 2, d(y) = y, then d is a (2, 1)-homogeneous k-derivation of degree 0. The zero derivation is γ-homogeneous of every degree. The sum of γ- homogeneous derivations of the same degree s is γ-homogeneous of degree s. If d 1, d 2 are γ-homogeneous derivations of degree s 1 and s 2, respectively, then the derivation [d 1, d 2 ] = d 1 d 2 d 2 d 1 is γ-homogeneous of degree s 1 +s 2. Proposition The following conditions are equivalent: (1) d is γ-homogeneous of degree s. (2) d(x i ) A (s+γ i) γ for i = 1,..., n. Proof. (1) (2). It is clear, because each x i belongs to A (γ i) γ. (2) (1). Every polynomial in A γ (p) is a sum of monomials of the form a α X α, where γα = p and a α k. Hence, if α Ω is such an element that γα = p, then it suffices to show that d(x α ) A γ (s+p). But d(x α ) = n i=1 α ix α 1 1 xα i 1 x αn n d(x i ), i α 1 γ (α i 1)γ i + + α n γ n = αγ γ i = p γ i, and d(x i ) A (s+γ i) γ. Therefore, d(x α ) A (p γ i) γ A (s+γ i) γ Corollary The derivation x i A (s+p) γ. is γ-homogeneous of degree γ i. A k-derivation d of k[x] is called monomial if d(x 1 ),..., d(x n ) are monomials. Proposition If d is a monomial k-derivation of k[x], then there exists a direction γ such that d is γ-homogeneous.

28 22 Part I. Preliminary concepts Proof. Assume that d(x i ) = a i X α i, for i = 1,..., n, where a i k and α i = (α i1,..., α in ) Ω. We must find a nonzero sequence (γ 1,..., γ n ) of integers and an integer s such that γ 1 α i1 + + γ n α in = γ i + s, for all i = 1,..., n. For this purpose consider the following system of n+1 linear equations over Q: (α 11 1)γ α 1n γ n + ( 1)s = 0 α 21 γ α 2n γ n + ( 1)s = 0 α n1 γ (α nn 1)γ n + ( 1)s = 0 0γ γ n + 0s = 0 Since the determinant of the main matrix of this system is equal to 0, there exists a nonzero solution (γ 1,..., γ n, s) belonging to Z n+1 and it is clear that (γ 1,..., γ n ) 0. Note also the following proposition which is easy to prove. Proposition Let d be a γ-homogeneous k-derivation of k[x] and let f k[x]. If f k[x] d, then each γ-homogeneous component of f belongs also to k[x] d. Corollary If d is a γ-homogeneous k-derivation of k[x], then k[x] d, the ring of constants with respect to d, is generated over k by γ-forms. In the present section we introduced several terms such as: γ- homogeneous, γ-form, γ-degree,.... If γ = (1,..., 1), then we have the ordinary direction and we omit the sign γ. So, in this case, we say: homogeneous, form, degree,.... Moreover, in this case, we write A (s) instead of A (s) γ. From Proposition we get Corollary The k-derivation d is homogeneous of degree s if and only if the polynomials d(x 1 ),..., d(x n ) are forms belonging to A (s+1). In particular, every linear k-derivation, that is, a k-derivation d of k[x] such that d(x i ) = n j=1 a ijx j for i = 1,..., n, is homogeneous of degree Darboux polynomials Let us introduce (as in [63], [64]) a new notion that dates back to Darboux s memoir [15]. Let d be a k-derivation of k[x]. We say that a

29 Chapter 2. Useful facts and preliminary results 23 polynomial f k[x] is a Darboux polynomial of d if f 0 and d(f) = hf, for some h k[x]. If f is a Darboux polynomial of d, then every h k[x], such that d(f) = hf, is said to be a polynomial eigenvalue of f. If k is a domain then such an h is unique. Darboux polynomials with nonzero eigenvalues (for k = R or C) are well known in the theory of polynomial differential equations. They coincide with the so-called partial first integrals (see, for example, [63] and [117]) of the system of polynomial differential equations determined by d. Every element belonging to the ring of constants with respect to d is of course a Darboux polynomial. In the vocabulary of differential algebra, Darboux polynomials coincide with generators of principal differential ideals, that is, f k[x] is a Darboux polynomial iff f 0 and the ideal (f) is differential (i. e., d(f) (f)). We know ([98], [95], [76]) that every differential ideal (in a noetherian case) has a differential primary decomposition. As a consequence of this fact we have the following proposition which, in our case, is easy to be proved. Proposition Assume that k is a UFD. If f k[x] is a Darboux polynomial of d, then all factors of f are also Darboux polynomials of d. Thus, looking for Darboux polynomials of a given k-derivation d (where k is a field) reduces to looking for irreducible ones. Note now some simple, but useful, propositions. Proposition Let d be a k-derivation of k(x) such that d(k[x]) k[x] (where k is a field). Let f and g be nonzero coprime polynomials in k[x]. Then f/g k(x) d iff f and g are Darboux polynomials with the same eigenvalue. Proposition Let γ = (γ 1,..., γ n ) be a direction and let d be a γ- homogeneous k-derivation of k[x] (where k is a domain). If f k[x] is a Darboux polynomial of d, then the eigenvalue h of f is a γ-form and all γ- components of f are also Darboux polynomials with the common eigenvalue equal to h. Proof. Compare γ-degrees of the equality d(f) = hf. See [64] for a detail. Note that, even for the ordinary direction, Darboux polynomials of a homogeneous derivation are not necessarily homogeneous. Indeed, let