Local monomialization of a system of first integrals of Darboux type

TL;DR

This paper proves that for singular foliations with first integrals of Darboux type, it is possible to locally transform the integrals into monomials, simplifying the structure of the foliation's singularities.

Contribution

It establishes the existence of local monomialization of first integrals and reduction of singularities for foliations generated by these integrals.

Findings

Existence of local monomialization of first integrals.

Reduction of singularities to monomial form.

Applicable to real and complex analytic foliations.

Abstract

Given a real- or complex-analytic singular foliation $\theta$ with $n$ first integrals of meromorphic or Darboux type $(f_1,\dots,f_n)$, we prove that there exists a local monomialization of the first integrals. In particular, if $\theta$ is generated by the $n$ first integrals, we prove the existence of a local reduction of singularities of $\theta$ to monomial singularities.