Positive Characteristic Darboux-Jouanolou Integrability of Differential Forms

TL;DR

This paper extends the Darboux-Jouanolou integrability theorem to polynomial differential forms over any field, providing new insights into algebraic integrability and methods for finding integrating factors.

Contribution

It generalizes the Darboux-Jouanolou theorem to arbitrary fields and explores Darboux's method for generating integrating factors for differential forms.

Findings

Proved a Darboux-Jouanolou type theorem for algebraic integrability over arbitrary fields

Investigated Darboux's method for producing integrating factors

Extended integrability results to polynomial differential forms in positive characteristic

Abstract

We prove a Darboux-Jouanolou type theorem on the algebraic integrability of polynomial differential $r$-forms over arbitrary fields ($r\geq 1$). We also investigate the Darboux's method for producing integrating factors.