Local monomialization conjecture of a singular foliation of Darboux type

TL;DR

This paper proposes a conjecture on local monomialization of Darboux type singular foliations, extending previous results and aiming to facilitate the study of pseudo-abelian integrals in complex dynamical systems.

Contribution

It introduces a new conjecture on monomialization for Darboux type foliations, expanding the theoretical framework for analyzing complex singular foliations.

Findings

Proposes a conjecture for local monomialization of Darboux type foliations.

Connects the conjecture to the study of pseudo-abelian integrals.

Builds upon previous work on monomialization of foliations.

Abstract

After the nice result introduced by Belotto in [1] concerning the local monomialization of a singular foliation given by n first integrals, this work is a continuation in the same spirit. In this paper, we introduce a important conjecture about local monomialization of a singular foliation of Darboux type (see section 1). This conjecture can be used to study pseudo-abelian integrals [2,4].