Polynomial derivations and their rings of constants

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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 ´N 1 9 9 4

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Polynomial derivations and their rings of constants i

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[x1, . . . , xn] over a field k of characteristic zero. The object of prin- cipal 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 f₁, . . . , f_n are polynomials belonging to k[X]. There exists then (see [8]) a unique k-derivation d of k[X] such that d(x₁) = f1, . . . , d(xn) = fn. The derivation d is defined by

d(h) = f₁ ∂h

∂x₁ + · · · + 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(x₁(t), . . . , x_n(t)), 1 6 i 6 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

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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(x1, . . . , xn) be the quotient field of k[X]. An element h of k[X] r k (resp. of k(X) r k) is said to be a polynomial (resp. rational) first integral of the system (0.2) if the following identity holds

f1

∂h

∂x₁ + · · · + fn

∂h

∂x_n = 0. (0.3)

Thus, the set of all the polynomial first integrals of (0.2) coincides with the set k[X]^drk 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)^drk, 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 com- puter 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 f1, . . . , fn 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 6= 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

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Introduction 3

ring of constants is not finitely generated (see Section 4.2). How to de- cide 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 small- est) 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[x²₁, . . . , 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 = \

d∈D

k[X]^d= {w ∈ k[X]; d(w) = 0 for all 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 4.1.4 describing all the subrings of a finitely generated k-algebra A (without zero divisors) which are rings of

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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 B₀∩ A = B, where B₀ 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 4.2.4 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]^dis 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 6 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 1988. 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 7.3.1 and 7.3.2.

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.

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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_kL <

∞ 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 3.2.7 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^dif 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^? =Pn

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 Derk(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 7.2.12) we prove that if d is a k-derivation of k[x, y], then k[x, y]^d 6= 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

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6 Introduction

function deg_d (Section 6.1) and by using its properties. A short proof of Theorem 8.1.1 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 The- orem 9.6.4 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 13.2.1. 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 Gr¨obner bases. With the help of this algorithm, using the CoCoA pro- gramme [3], we give (in Sections 6.8 and 6.9) a series of examples related to the finite generating sets of the Weitzenb¨ock 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.

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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 N0 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 A₀ its field of fractions.

If (i1, . . . , is) is a sequence of nonnegative integers, then we denote by hi₁, . . . , i_si the Newton number (i₁+ · · · + i_s)!(i₁! · · · 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, d₁, d₂ ∈ Der(R) and x ∈ R, then the mappings xd, d1 + d2 and [d1, d2] = d1d2− d₂d1 are also derivations. Thus, the set Der(R) is an R-module which is also a Lie algebra. Note the following simple propositions

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8 Part I. Preliminary concepts

Proposition 1.1.1. If d is a derivation of R, x1, . . . , xs ∈ R and n > 0, then

dⁿ(x₁· · · x_s) = X

i1+···+is=n

hi₁, . . . , i_sidⁱ¹(x₁) · · · dⁱ^s(x_s).

Proposition 1.1.2. Let d₁, d₂ be derivations of R and let A = {x ∈ R; d₁(x) = d₂(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 (R₁, d₁) and (R₂, d₂) be differential k-algebras and let f : R₁ −→ R₂ be a homomorphism of k-algebras. Put T (f ) = {x ∈ R₁; f d1(x) = d2f (x)}. The homomorphism f is called differential if f d₁ = d₂f , i. e., if T (f ) = R₁. The set T (f ) is a k-subalgebra of R₁. As a consequence of this fact we get

Proposition 1.1.3. If R1 = k[a1, . . . , as] is finitely generated over k, then f is differential iff a1, . . . , as∈ 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 RS be the k- algebra of fractions of R with respect to S. Then there exists a unique k-derivation d_S of R_Ssuch that the natural homomorphism R −→ R_S, r 7→

r/1, is differential. The derivation dS is defined by the formula: dS(r/s) = (d(r)s − rd(s))/s², 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 =T

d∈DR^d.

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Chapter 1. Definitions, notations and basic facts 9

1.2 Derivations in polynomial rings

Let R[X] = R[xi; 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(xj) = 0 for all j 6= 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(xi) = 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 (i1)_∂x^∂w

i1 + · · · f (in)_∂x^∂w

in,

where {i1, . . . , in} is a finite subset of I such that w ∈ R[x_i₁, . . . , xin]. As a consequence of the above facts we get

Theorem 1.2.1. Let k[X] = k[x₁, . . . , x_n] be a polynomial ring over a ring k.

(1) If f₁, . . . , f_n ∈ k[X], then there exists a unique k-derivation d of k[X] such that d(x₁) = f₁, . . . , d(x_n) = f_n. This derivation is of the form:

d = f1 ∂

∂x1 + · · · + fn ∂

∂xn.

(2) Der_k(k[X]) is a free k[X]-module on the basis _∂x^∂

1, . . . ,_∂x^∂

n. (3) _∂x^∂

i

∂

∂xj = _∂x^∂

j

∂

∂xi, for all i, j ∈ {1, . . . , n}.

(4) If d ∈ Der_k(k[X]) and f ∈ k[X], then d(f ) =Pn i=1

∂f

∂xid(x_i). Note also the following

Proposition 1.2.2. Let d be a k-derivation of a k-algebra A. Let f ∈ k[X] = k[x₁, . . . , x_n] and a = (a₁, . . . , a_n) ∈ Aⁿ. Then f (a) is an element of A and d(f (a)) =Pn

i=1

∂f

∂xid(ai).

Proof. Put M = {g ∈ k[X]; d(g(a)) = _∂x^∂g

1d(a1) + · · · + _∂x^∂g

nd(an)}.

It is easy to check that M is a k-subalgebra of k[X] containing x₁, . . . , 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

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Proposition 1.2.3. If k is Z-torsion free and k[t, x1, . . . , xn] is the polynomial ring, then k[t, x₁, . . . , x_n]^d= k[x₁, . . . , x_n], where d = _∂t^∂.

By the above proposition and Proposition 1.1.3, we get

Proposition 1.2.4. Assume that k is Z-torsion free. Let A = k[t, x₁, . . . , x_n], B = k[u, y₁, . . . , y_m] be the polynomial rings over k. Put dA= _∂t^∂, dB = _∂u^∂ and let f : A −→ B be a homomorphism of k-algebras.

Then the following conditions are equivalent:

(1) f is differential (that is, f d_A= d_Bf ).

(2) The polynomials f (t) − u, f (x1), . . . , f (xn) belong to k[y1, . . . , ym].

Proof. (1) ⇒ (2). In view of Proposition 1.2.3 we know, that k[y1, . . . , ym] = B^d^B. So, we must show that dB(f (t) − u) = dB(f (x1)) =

· · · = d_B(f (x_n)) = 0. If i ∈ {1, . . . , n}, then d_B(f (x_i)) = f d_A(x_i) = f (0) = 0. Moreover, dB(f (t) − u) = f dA(t) − dB(u) = f (1) − 1 = 0.

(2) ⇒ (1). The variables t, x₁, . . . , x_n belong to T (f ) = {a ∈ A; f d_A(a) = d_Bf (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 pre- vious section, one can prove analogous properties for the fields of rational functions. In particular, we have the following three propositions.

Proposition 1.3.1. 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 1.3.2. Let k(X) = k(x₁, . . . , x_n).

(1) If f₁, . . . , f_n ∈ k(X), then there exists a unique k-derivation d of k(X) such that d(x1) = f1, . . . , d(xn) = fn. This derivation is of the form:

d = f₁_∂x^∂

1 + · · · + f_n_∂x^∂

n. (2) The derivations _∂x^∂

1, . . . ,_∂x^∂

n form a basis of the k(X)-space Der_k(k(X)).

(3) _∂x^∂

i

∂

∂xj = _∂x^∂

j

∂

∂xi for all i, j ∈ {1, . . . , n}.

(4) If d ∈ Derk(k(X)) and f ∈ k(X), then d(f ) =Pn i=1

∂f

∂xid(xi).

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Proposition 1.3.3. Let k ⊆ L be fields. Let d be a k-derivation of L and let a = (a₁, . . . , a_n) ∈ Lⁿ. If F = f /g is an element of k(x₁, . . . , x_n) such that g(a) 6= 0, then F (a) is an element of L and

d(F (a)) = ∂F

∂x₁d(a₁) + · · · + ∂F

∂x_nd(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 1.3.4. Suppose that k is a field of characteristic zero and let d = _∂t^∂. Then k(t, x1, . . . , xn)^d = k(x1, . . . , xn) and k(t)[x1, . . . , xn]^d = k[x1, . . . , xn].

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 1.4.1. 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 1.4.2. Let L = k(p₁, . . . , p_n) be a finitely generated extension of a field k of characteristic zero. Put p = (p1, . . . , pn) and let Mp = {f ∈ k[x₁, . . . , x_n]; f (p) = 0}. The following conditions are equivalent:

(1) L is algebraic over k.

(2) There exist n polynomials f1, . . . fn∈ M_p such that det [_∂x^∂fⁱ

j(p)] 6= 0.

Proposition 1.4.3. Let L = k(x1, . . . , xn) be the field of rational functions over a field k of characteristic zero. If f₁, . . . , f_n ∈ L, then the following

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conditions are equivalent:

(1) The elements f₁, . . . , f_n are algebraically independent over k.

(2) If d is a k(f₁, . . . , f_n)-derivation of L, then d = 0.

(3) det [_∂x^∂fⁱ

j] 6= 0.

1.5 Derivations in power series rings

Let k[[T ]] = k[[t1, . . . , tn]] be the power series ring over a ring k in n variables and let M denote the ideal (t1, . . . , tn). Let us recall that k[[T ]] is a complete ring with respect to the M -adic topology. SinceT∞

m=0M^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(t1) = · · · = d(tn) = 0, then d = 0.

Using the above facts and the same arguments as in Section 1.2, we get Proposition 1.5.1. Let d be a derivation of k and let f₁, . . . , f_n ∈ k[[T ]].

Then there exists a unique derivation D of k[[T ]] such that D|k = d and D(t1) = f1, . . . , D(tn) = fn. The derivation D is defined by the formula:

D = ˜d + f₁_∂t^∂

1 + · · · + f_n_∂t^∂

n. Theorem 1.5.2 ([8]).

(1) If f1, . . . , fn ∈ k[[T ]], then there exists a unique k-derivation d of k[[T ]] such that d(t₁) = f₁, . . . , d(t_n) = f_n. The derivation d is of the form:

d = f₁_∂t^∂

1 + · · · + 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

∂

∂tj = _∂t^∂

j

∂

∂ti, for all i, j ∈ {1, . . . , n}.

(4) If d is a k-derivation of k[[T ]], then d(f ) = _∂t^∂f

1d(t1) + · · · +_∂t^∂f

nd(tn), for every f ∈ k[[T ]].

Now we will explain two differential formulas concerning rings of power series.

Denote by Ω = Ωn the set {α = (α1, . . . , αn); α1, . . . , αn ∈ N0}. If α = (α₁, . . . , α_n) is an element of Ω, then we denote by T^α the monomial t^α₁¹· · · t^α_nⁿ. In particular, T⁰ = t⁰₁· · · t⁰_n = 1 and T^αT^β = T^α+β for any α, β ∈ Ω. Every element of k[[T ]] has a unique decomposition of the form P

αa_αT^α, where a_α∈ k for all α ∈ Ω.

If α = (α1, . . . , αn) ∈ Ω, then |α| denotes the sum α1+· · ·+αn. The ideal M^m is generated by the set {T^α; |α| = m}. If m ∈ N andP a_αT^α∈ M^m, then a_α = 0, for all α such that |α| < m.

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Put R = k[[T ]] and consider a second power series ring R[[Y ]] = R[[y₁, . . . , y_s]] = k[[t₁, . . . , t_n, y₁, . . . , y_s]]. Let f = P

β∈Ωsf_βY^β = P

β∈Ωsf_βy₁^β¹· · · y_s^β^s be a series in R[[Y ]] and let ϕ₁, . . . , ϕ_s be series, without constant terms, belonging to R. If p ∈ N, then put Cp =P

|β|<pf_βϕ^β = P

|β|<pf_βϕ^β₁¹· · · ϕ_s^β^s. It is clear that C_p∈ R. Since ϕ₁, . . . , ϕ_shave no constant terms, Cp+1− C_p∈ M^p. This implies that (Cp) is a Cauchy sequence in R and hence, this sequence is convergent. We denote byP

β∈Ωsf_βϕ^β or by f (ϕ) = f (ϕ₁, . . . , ϕ_s) the limit of (C_p).

Now we may state the following power series analogy of Proposi- tion 1.2.2.

Proposition 1.5.3. Let d be a derivation of R = k[[T ]], let f ∈ R[[Y ]], and let ϕ₁, . . . , ϕ_s be series in R without constant terms. Put ϕ = (ϕ₁, . . . , ϕ_s).

Then f (ϕ) is an element of R and

d(f (ϕ)) = ˜d(f )(ϕ) +Ps i=1

∂f

∂yi(ϕ)d(ϕi), (1.1) where ˜d is the derivation of R[[Y ]] defined by ˜d(P

βf_βY^β) =P

βd(f_β)Y^β. Proof. We already know that f (ϕ), _∂y^∂f

1(ϕ), . . . ,_∂y^∂f

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) −Ps i=1

∂f

∂yi(ϕ)d(ϕi) (1.2) belongs to M^p, for any p ∈ N. Let p ∈ N and let f =P

βfβY^β. Set Ep = X

|β|>p

fβY^β, Ep(ϕ) = X

|β|>p

fβϕ^β,

Fp = X

|β|6p

f_βY^β, Fp(ϕ) = X

|β|6p

f_βϕ^β.

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 ^∂E_∂y^p

i(ϕ) belong to M^p, for every i = 1, . . . , s. Since F_p is a polynomial in R[y₁, . . . , y_s], we have (see Section 1.2):

d(F_p(ϕ)) − ˜d(F_p)(ϕ) −Ps i=1

∂Fp

∂yi(ϕ)d(ϕ_i) = 0.

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Therefore, the element A, defined by (1.2), is equal to d(Ep(ϕ)) − ˜d(Ep)(ϕ) −Ps

i=1

∂Ep

∂yi(ϕ)d(ϕi), and so, it is an element of M^p.

Corollary 1.5.4. Let d be a k-derivation of k[[T ]], let f ∈ k[[y1, . . . , ys]]

and let ϕ = (ϕ₁, . . . , ϕ_s) ∈ k[[T ]]^s. Assume that the series ϕ₁, . . . , ϕ_s have no constant terms. Then f (ϕ) ∈ k[[T ]] and

d(f (ϕ)) =

s

X

i=1

∂f

∂y_i(ϕ)d(ϕ_i).

Assume now that {B_α}_α∈Ωis a family of elements of k[[T ]]. Then there exists an element of k[[T ]] of the form P

α∈ΩBαT^α. This element is the limit of the Cauchy sequence (C_p), where C_p = P

|α|6pB_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 formP

α∈Ωd(Bα)T^α. Moreover, we have also the elementP

αB_αd(T^α).

Proposition 1.5.5. d(P

αBαT^α) =P

αd(Bα)T^α+P

αBαd(T^α).

Proof. Use the same argument as in the proof of Proposition 1.5.3.

1.6 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[x1, . . . , 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 1.6.1. Let f₁, . . . , f_n∈ k[X][[t]] and a₁, . . . , a_n∈ k. Then there exist unique series ϕ1, . . . , ϕn∈ k[[t]] such that:

(1) ^∂ϕ_∂tⁱ = fi(ϕ1, . . . , ϕn), for i = 1, . . . , n, and

(2) the constant terms of ϕ₁, . . . , ϕ_n are equal to a₁, . . . , a_n, respectively.

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For the proof of this theorem we need some notations and lemmas.

Assume that B₀, B₁, . . . is a sequence of series belonging to k[[t]] and put Cp =Pp

i=1Bitⁱ for any p ∈ N0. Since Cp+1− C_p∈ (t)^p, these series form a Cauchy sequence in k[[t]] and hence, there exists the series P∞

p=0B_pt^p. It is easy to check the following

Lemma 1.6.2. If Bp = P∞

i=0bpitⁱ with bpj ∈ k, then P∞

p=0Bpt^p = P∞

s=0ast^s, where as=P

i+j=sbij for any s ∈ N0.

Let f be a series in k[X][[t]] and let ϕ1, . . . , ϕn ∈ k[[t]]. Set ϕ = (ϕ₁, . . . , ϕ_n) and let f =P∞

p=0f_pt^p, where each f_p is a polynomial in k[X].

Then we have the sequence f₀(ϕ), f₁(ϕ), . . . of series in k[[t]] and so, we can form the series f (ϕ) =P∞

p=0fp(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 1.6.3. If g is a nonzero polynomial in k[X], then there exists a point a ∈ kⁿ such that g(a) 6= 0.

Proof. Let us recall that k[X] = k[x1, . . . , xn] and Q ⊆ k.

First assume that n = 1. Set x = x₁ and let g =Pp

j=0g_jx^j with g_j ∈ k and gp 6= 0. If p = 0 then our lemma is evident. Let p > 0 and suppose that g(a) = 0 for all a ∈ k¹. Put h(x) = 2^pg(x) − g(2x). Then h(a) = 0 (for all a ∈ k¹) and deg h < deg g. So, by induction, h = 0. This implies that g0 = · · · = gp−1 = 0 and we have a contradiction: 0 = g(1) = gp 6= 0.

This completes the proof for n = 1.

Assume now that n > 1 and let g =Pp

j=0gjx^jnwith gj ∈ k[x₁, . . . , xn−1] and gp 6= 0. Then, by induction, there exists b ∈ kⁿ⁻¹ such that gp(b) 6= 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, an) 6= 0, for some an∈ k.

As a consequence of the above lemma we get

Lemma 1.6.4. Let f, g ∈ k[X][[t]]. If f (a) = g(a) for any a ∈ kⁿ, then f = g.

If ψ = P∞

i=0aitⁱ ∈ k[[t]] and p ∈ N0, then denote by ψ_[p] the p-th coefficient of ψ (that is, ψ_[p] = ap). Moreover, denote by sp(ψ) the sum Pp

i=0a_itⁱ.

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Lemma 1.6.5. Let f ∈ k[X][[t]], p ∈ N0 and ϕ1, . . . , ϕn ∈ k[[t]]. Then f (ϕ₁, . . . , ϕ_n)_[p] = f (s_p(ϕ₁), . . . , s_p(ϕ_n))_[p].

Proof. Observe that if ϕ and ψ are two series in k[[t]], then (ϕ+ψ)_[p]= (sp(ϕ) + sp(ψ))_[p] and (ϕ · ψ)_[p] = (sp(ϕ) · cp(ψ))_[p]. So Lemma holds for f ∈ k[X]. The rest follows from Lemma 1.6.2.

Proof of Theorem 1.6.1. We construct the coefficients (ϕ1)_[p], . . . , (ϕn)_[p]

using induction on p. If p = 0 then put

(ϕ₁)_[0] = a₁, . . . , (ϕ_n)_[0] = a_n. (1.3) Let p > 0 and assume that, for all j 6 p, the coefficients (ϕ1)_[j], . . . , (ϕ_n)_[j]

are already constructed. If i ∈ {1, . . . , n}, then we define:

(ϕi)_[p+1]= 1

p + 1fi(g1, . . . , gn)_[p], where gi=

p

X

j=0

(ϕi)_[j]t^j. (1.4)

Thus, we have constructed the series ϕ1, . . . , ϕn ∈ k[[t]] (where ϕ_i = P∞

p=0(ϕi)_[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 = (ϕ₁, . . . , ϕ_n) is an arbitrary sequence of series in k[[t]] satisfying (1) and (2). Then (^∂ϕ_∂tⁱ)_[p] = fi(ϕ1, . . . , ϕn)_[p], for any p ∈ N0 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 = (f₁, . . . , f_n) is a sequence of series in k[X][[t]] and a = (a₁, . . . , a_n) ∈ kⁿ, then denote by ϕ(t, a) = (ϕ₁(t, a), . . . , ϕ_n(t, a)) the unique sequence (ϕ1, . . . , ϕn) of series from k[[t]] satisfying the conditions (1) and (2) of Theorem 1.6.1. The sequence ϕ(t, a) is called the formal solution of the differential system

∂X

∂t = f (X), X_[0] = a. (1.5)

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Now we prove the following polynomial property of the formal solutions.

Theorem 1.6.6. If f ∈ k[X][[t]]ⁿ, then there exist uniquely determined polynomials

ωij ∈ k[X] (for i ∈ {1, . . . , n} and j ∈ N0) such that

ϕi(t, a) =

∞

X

j=0

ωij(a)tⁿ, (1.6)

for all a ∈ kⁿ and i ∈ {1, . . . , n}, where ϕ(t, a) is the formal solution of (1.5).

Proof. Similarly as in the proof of Theorem 1.6.1 we construct (using an induction on p ∈ N0) the polynomials ω_1p, ω_2p, . . . , ω_np. If p = 0 then put ω01= x1, . . . , ω0n= xn.

Let p > 0 and assume that, for all j 6 p, the polynomials ω1j, . . . , ωnj

are already defined. Then define:

ω_i(p+1) = 1

p + 1f_i(G₁, . . . , G_n)_[p], where G_i=Pp

j=0ω_ijt^j for any i = 1, . . . , n. Now, using Lemma 1.6.5, one can easily deduce that the polynomials ωij satisfy (1.6). The uniqueness follows from Lemma 1.6.3 and Theorem 1.6.1.

If a = (a₁, . . . , a_n) ∈ kⁿ, then we denote by π_athe surjective homomorphism from k[X][[t]] to k[[t]] defined by π_a(f ) = f (a).

Corollary 1.6.7. If f ∈ k[X][[t]]ⁿ, then there exist unique series W₁, . . . , W_n ∈ k[X][[t]] such that ϕ_i(t, a) = π_a(W_i) for all a ∈ kⁿ and i ∈ {1, . . . , n}.

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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[x₁, . . . , x_n] and k(X) = k(x₁, . . . , x_n) for the ring of polynomials and the field of rational functions, respectively. By a direction we mean a nonzero sequence γ = (γ1, . . . , γn) 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(x1), . . . , d(xn) 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^?=Pn

i=1∂d(xi)/∂xi. 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)) 6= k(X). We obtain it as a consequence

18

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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_kL < ∞, then d(L) 6= 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 = (γ₁, . . . , γ_n) be a direction and let s ∈ Z. If α = (α1, . . . , α_n) ∈ Ω = Nⁿ₀, then γα denotes the sum γ₁α₁+ · · · + γ_nα_n and X^α denotes the monomial x^α₁¹· · · x^α_nⁿ.

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 =P

γα=sa_αX^α, where aα ∈ 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 y⁵ + xy³ − 6x²y is a (2, 1)-form of degree 5, x⁴y + x⁷y³ is a (−2, 3)-form of degree −5 and x⁸y⁴+ x⁶y³+ 7 is a (1, −2)-form of degree 0.

Proposition 2.1.1. Assume that k is a domain of characteristic zero and k0 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^γ¹x1, . . . , t^γⁿxn) = t^sf (x1, . . . , xn) (in the ring k0(t)[X]).

(3) γ1x1 ∂f

∂x1 + · · · + γnxn ∂f

∂xn = sf .

Proof. (3) ⇒ (2). Set R = k₀(t)[X] = k₀(t)[x₁, . . . , x_n], u = (t^γ¹x1, . . . , t^γⁿxn), g = t^−sf (u), δ = _∂t^∂, and let ϕ : k0[X] −→ R be the homomorphism of k0-algebras defined by ϕ(xi) = t^γⁱxi, for i = 1, . . . , n.

Applying ϕ for (3) we get γ₁t^γ¹x₁_∂x^∂f

1(u) + · · · + γ_nt^γⁿx_n_∂x^∂f

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 k₀[X]. Thus, we have the equality:

f (t^γ¹x₁, . . . , t^γⁿx_n) = f (u) = t^sg(x₁, . . . , x_n).

Substituting now t = 1, we get: g(x₁, . . . , x_n) = f (x₁, . . . , x_n) and hence, we have (2).

The implications (2) ⇒ (1) and (1) ⇒ (3) are well known and easy to be proved.

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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 is a k-submodule of k[X] and k[X] =L

s∈ZA^(s)_γ . Moreover, A^(s)_γ A^(t)_γ ⊆ 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 =P f_s into γ-components f_s of degree s. If f 6= 0, then γ-deg(f ) denotes the γ-degree of f , that is, the maximal s such that fs6= 0. We assume also that

γ-deg(0) = −∞.

It is easy to check the following

Lemma 2.1.2. Let k be a domain and let f, g be nonzero polynomials in k[X]. If f g 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, x₁, . . . , x_n), the following equality holds:

ϕ(t^γ¹x1, . . . , t^γⁿxn) = t^sϕ(x1, . . . , xn).

Proposition 2.1.3. 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.

(3) γ1x1∂ϕ

∂x1 + · · · + γnxn ∂ϕ

∂xn = sϕ.

Proof. (1) ⇒ (2). Consider the γ-decompositions f = f_p₁+· · ·+f_p, g = gq1 + · · · + gq, where p1 < · · · < p, q1 < · · · < q, and put X = (x1, . . . , xn), t^γX = (t^γ¹x₁, . . . , t^γⁿx_n).

By the equality ϕ(t^γX) = t^sϕ(X) we obtain the following equality of polynomials in k(t)[X]:

f (t^γX)g(X) = t^sf (X)g(t^γX).

Hence, by Proposition 2.1.1,

(t^p¹f_p₁(X) + · · · + t^pf_p(X))g(X) = t^sf (X)(t^q¹g_q₁(X) + · · · + t^qg_q(X)).

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Chapter 2. Useful facts and preliminary results 21

Comparing now powers of t we see that p = s + q and fp(X)g(X) = g_q(X)f (X). Since the polynomials f and g are coprime, f_p = f h 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 2.1.1.

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 + y², 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 d1, d2 are γ-homogeneous derivations of degree s1 and s2, respectively, then the derivation [d₁, d₂] = d₁d₂−d₂d₁is γ-homogeneous of degree s₁+s₂. Proposition 2.1.4. The following conditions are equivalent:

(1) d is γ-homogeneous of degree s.

(2) d(xi) ∈ A^(s+γγ ⁱ⁾ for i = 1, . . . , n.

Proof. (1) ⇒ (2). It is clear, because each x_i belongs to A^(γγⁱ⁾.

(2) ⇒ (1). Every polynomial in A^(p)γ is a sum of monomials of the form a_αX^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^(s+p)γ . But d(X^α) =Pn

i=1αix^α₁¹· · · x_i^αⁱ⁻¹· · · x^α_nⁿd(xi),

α₁γ₁+ · · · + (α_i− 1)γ_i+ · · · + α_nγ_n= αγ − γ_i = p − γ_i, and d(xi) ∈ A^(s+γγ ⁱ⁾. Therefore, d(X^α) ∈ Aγ^(p−γⁱ⁾A^(s+γγ ⁱ⁾⊆ A^(s+p)_γ . Corollary 2.1.5. The derivation _∂x^∂

i is γ-homogeneous of degree −γi. A k-derivation d of k[X] is called monomial if d(x1), . . . , d(xn) are monomials.

Proposition 2.1.6. If d is a monomial k-derivation of k[X], then there exists a direction γ such that d is γ-homogeneous.

Polynomial derivations and their rings of constants

Andrzej Nowicki