Andrzej Nowicki

Around Jouanolou non-integrability theorem☆

Abstract We generalize the well-known Jouanolou non-integrability theorem concerning the system of ordinary differential equations: x ′ 1 = x s 2 , x ′ 2 = x s 3 , x ′ 3 = x s 1 , s ϵ N , s ϵ N , s ≥ 2 to an arbitrary prime number n ≥ 3 of variables and arbitrary integer exponent s ≥ 3. Our proof is completely elementary.

Rings and fields of constants for derivations in characteristic zero

Abstract Let k be a field of characteristic zero and A a finitely generated k -algebra. We give a description of all k -subalgebras of A which are rings of constants for derivations of A . Moreover we show some applications of our description.

SIMPLE QUADRATIC DERIVATIONS IN TWO VARIABLES

Let k[x, y] be the polynomial ring in two variables over an algebraically closed field k of characteristic zero. We call quadratic derivations the derivations of k[x, y] of the form where a(x), b(x) ∈ k[x]. We are interested in simple derivations of this type; every such derivation is equivalent to Δ p = ∂/∂x + (y 2 − p(x))∂/∂y for a suitable p in k[x]. For some p, we are able to decide the simplicity of Δ p : if the degree of p is odd, then Δ p is simple; if p has degree 2, then Δ p is simple if and only if p fulfills an arithmetic condition. *Supported by KBN Grant 2 PO3A 017 16.

A note on locally nilpotent derivations

Let R be a commutative reduced, Z-torsion free ring. Let d and δ be two locally nilpotent derivations of R which commute, a an element of R. We prove that the derivation ad + δ is locally nilpotent if and only if d(a) = 0.

The Lie structure of a commutative ring with a derivation

Ro], for all n > 0. Let A be a non-zero Lie ideal of R o and assume that R is 2-torsion free. In this situation C.R. Jordan and D.A. Jordan ([3], [4]) proved the following four theorems: (1) IfR is prime and A = R o or A = R 1 then A is a prime Lie ring ([3] Theorem 6, [4] Theorems I and 4). (2) If R is noetherian d-prime and A = R o or A = R 1 then A is a prime Lie ring ([3] Theorem 7, [4] Theorems 2 and 5). (3) If R is noetherian d-prime and A = R o or A = R 1 then every non-zero Lie ideal of A contains a non-zero d-ideal of R ([3] Theorem 8, [4] Theorems 3 and 6). [4] If R is noetherian then R o is simple if and only if R is d-simple ([4] Theorem 3). In this paper we show that Theorems (1)-(3) are also true in the case where A =

Generators of rings of constants for some diagonal derivations in polynomial rings

Abstract Let K be a field of characteristic zero. We show that if n ⩾ 3, given r ⩾ 0 there exists a diagonal K -derivation of K [ x 1 , …, x n ] such that the minimal number of generators over K of the ring of constants is equal to r .

On the Jacobian conjecture in two variables

Let k be a field of characteristic zero. The two-dimensional Jacobian conjecture states that given ƒ and g in k[x,y], if the Jacobian of (ƒ,g) is a non-zero constant, then k[ƒ,g] = k[x,y]. In this paper we give new proofs of known equivalent versions of this conjecture.

Rings of Constants of the Form k[f]

Let k[X] be the algebra of polynomials in n variables over a field k of characteristic zero, and let f ∈ k[X]∖ k. We present a construction of a derivation d of k[X] whose ring of constants is equal to the integral closure of k[f] in k[X]. A similar construction for fields of rational functions is also given.

COMMUTATIVE BASES OF DERIVATIONS IN POLYNOMIAL AND POWER SERIES RINGS

In this note we give at first a description of commutative basis of modules Derk(k[xl, ..., xn]) and Derk(kI[xl, ..., xn]]) in characteristic zero (Theorem 2), and next we prove, using a theorem of Nousiainen and Sweedler [3, Theorem 3.3], an equivalent version of the Jacobian Conjecture (Theorem 5). Let k be a commutative ring containing the field Q of rational numbers and let R denote either the ring k[Xl,... ,xn] of polynomials over k or the ring k[[Xl, ..., Xn]] of formal power series over k. We denote by A 1, ..., An the partial derivatives O/Oxl, ..., O/Oxn, and by Derk(R) the R-module of all k-derivations from R to R. It is well-known [2] that Derk(R) is a free R-module on the basis A~, . . . ,An.

Generic polynomial vector fields are not integrable

Abstract We study some generic aspects of polynomial vector fields or polynomial derivations with respect to their integration. In particular, using a well-suited presentation of Darboux polynomials at some Darboux point as power series in local Darboux coordinates, it is possible to show, by algebraic means only, that the Jouanolou derivation in four variables has no polynomial first integral for any integer value s ≥ 2 of the parameter. Using direct sums of derivations together with our previous results we show that, for all n ≥ 3 and s ≥ 2, the absence of polynomial first integrals, or even of Darboux polynomials, is generic for homogeneous polynomial vector fields of degree s in n variables.