Darboux theory of integrability for real polynomial vector fields on the $n-$dimensional ellipsoid

TL;DR

This paper extends Darboux's integrability theory to polynomial vector fields on n-dimensional ellipsoids, providing new bounds on invariant structures and generalizing previous results from spheres and hyperplanes.

Contribution

It introduces a generalized Darboux theory for n-dimensional ellipsoids and establishes bounds on invariant parallels and meridians based on polynomial degree.

Findings

Maximum number of invariant parallels and meridians determined

Extension of bounds from hyperplanes to ellipsoids

Generalization of Darboux integrability to higher dimensions

Abstract

We extend to the $n$-dimensional ellipsoid contained in $\R^{n+1},$ the Darboux theory of integrability for polynomial vector fields in the $n$-dimensional sphere (Llibre et al., 2018). New results on the maximum number of invariant parallels and meridians of polynomial vector fields $\X$ on the invariant $n-$dimensional ellipsoid, as a function of its degree, are provided. Our results extend the known result on the upper bound for the number of invariant hyperplanes that a polynomial vector field $\Y$ in $\R^n$ can have in function of the degree of $\Y$.