Darboux theory of integrability in the sparse case

TL;DR

This paper extends Darboux's and Jouanolou's theorems to polynomial vector fields by using the Newton polytope's size, providing optimal bounds for the existence of rational first integrals.

Contribution

It introduces a new approach based on Newton polytopes to establish bounds for integrability, generalizing classical results in polynomial vector field theory.

Findings

Bounds are expressed in terms of Newton polytope size

Results are shown to be optimal

Extends classical theorems to the sparse case

Abstract

Darboux's theorem and Jouanolou's theorem deal with the existence of first integrals and rational first integrals of a polynomial vector field. These results are given in terms of the degree of the polynomial vector field. Here we show that we can get the same kind of results if we consider the size of a Newton polytope associated to the vector field. Furthermore, we show that in this context the bound is optimal.