Derivations of polynomial algebras without Darboux polynomials

Abstract

We present several new examples of homogeneous derivations of a polynomial ring k[X ] = k[x 1 , . . . , x n ] over a field k of characteristic zero without Darboux polynomials. Using a modification of a result of Shamsuddin, we produce these examples by induction on the number n of variables, thus more easily than the previously known example multidimensional Jouanolou systems ofŻoładek.

Finding non-polynomial positive invariants and lyapunov functions for polynomial systems through Darboux polynomials

“…Darboux polynomials have been widely studied as a basis of algorithmic procedures for integrating ODEs [1], and deriving first integrals of ODEs [16]. Jouanolou [17] constructs a polynomial vector field for which no Darboux polynomial exist; this work has been extended later by Ollagnier and Nowicki [18]. It is also well-known that there is no apriori bound on the degree of the Darboux polynomials in terms of the degree of the RHS of the ODEs.…”

Rings of constants of linear derivations on Fermat rings

“…, n. It is well known the difficulty to describe the ring of constants of an arbitrary derivation (see [1,3,4,5]). It is also difficult to decide if the ring of constants of a derivation is trivial (see [5,6,7]). In this work we study the ring of constants of linear derivations of Fermat rings and its locally nilpotent derivations.…”

A Note on Darboux Polynomials of Monomial Derivations

“…In [7], there is a full description of all monomial derivations of k[x, y, z] with trivial field of constants. Using this description and several additional facts, Moulin-Ollagnier and Nowicki present full lists of homogeneous monomial derivations of degrees s ≤ 4 (of k[x, y, z]) without Darboux polynomials in [8] and then in [9], they prove that a monomial derivation d (of k[x, y, z]) has no Darboux polynomials if and only if d has a trivial field of constants and x i ∤ d(x i ) for all i = 1, . .…”