# Darboux theory of integrability for real polynomial vector fields on $\sss^n$

## TL;DR

This paper surveys Darboux integrability for polynomial vector fields on Euclidean space and spheres, and introduces new bounds on the number of parallels and meridians based on the vector field's degree.

## Contribution

It extends Darboux theory to spheres and provides new maximum bounds for parallels and meridians of polynomial vector fields on $ ext{S}^n$.

## Key findings

- New bounds on parallels and meridians depending on degree
- Extension of Darboux integrability theory to spheres
- Comparison with hyperplanes in $ ext{R}^n$

## Abstract

This is a survey on the Darboux theory of integrability for polynomial vector fields, first in $\R^n$ and second in the $n$-dimensional sphere $\sss^n$. We also provide new results about the maximum number of parallels and meridians that a polynomial vector field $\X$ on $\sss^n$ can have in function of its degree. These results in some sense extend the known result on the maximum number of hyperplanes that a polynomial vector field $\Y$ in $\R^n$ can have in function of the degree of $\Y$.