Liouvillian integrability of vector fields in higher dimensions

TL;DR

This paper extends classical results on Liouvillian integrability from two-dimensional vector fields to higher dimensions, providing a new proof technique and characterizing integrals as successive integrations over algebraic extensions.

Contribution

It generalizes Singer's classical theorem to higher dimensions and introduces a novel Puiseux series method for simplifying integrability proofs.

Findings

Existence of Liouvillian first integrals in higher dimensions

Successive integrations over algebraic extensions characterize integrability

Elementary proofs of classical theorems for rational forms and vector fields

Abstract

We consider complex rational vector fields in dimension $n>2$ (equivalently, differential forms of degree $n-1$ in $n$ variables) which admit a Liouvillian first integral. Extending a classical result by Singer for $n=2$, our main result states that there exists a first integral which is obtained by two successive integrations from one-forms with coefficients in a finite algebraic extension of the rational function field. The proof uses Puiseux series in a novel way to simplify computations. We also apply this method to give elementary proofs of Singer's theorem for rational one-forms, and of the Prelle-Singer theorem on elementary integrability of rational vector fields.