A Note on Darboux Polynomials of Monomial Derivations

TL;DR

This paper investigates a specific monomial derivation in four variables, establishing a precise condition linking the absence of Darboux polynomials to the triviality of its field of constants.

Contribution

It proves that the monomial derivation has no Darboux polynomials if and only if its field of constants is trivial, clarifying the relationship between these properties.

Findings

No Darboux polynomials exist if and only if the field of constants is trivial.

Provides a characterization of the monomial derivation's properties.

Enhances understanding of derivations in polynomial rings.

Abstract

We study a monomial derivation $d$ proposed by J. Moulin Ollagnier and A. Nowicki in the polynomial ring of four variables, and prove that $d$ has no Darboux polynomials if and only if $d$ has a trivial field of constants.