On the number of Puiseux exponents of an invariant branch of a vector field

TL;DR

This paper establishes that the multiplicity of a plane analytic 1-form bounds the number of Puiseux exponents of an invariant branch, regardless of whether the foliation is dicritical.

Contribution

It proves a universal bound on Puiseux exponents based on the multiplicity of the 1-form, applicable to both dicritical and non-dicritical foliations.

Findings

Multiplicity bounds the number of Puiseux exponents

Applicable to formal and convergent branches

Valid for dicritical and non-dicritical cases

Abstract

We show that the multiplicity of a plane analytic 1-form is a bound for the number of Puiseux exponents of a (formal or convergent) branch. This is true whether the associated foliation is dicritical or not.