On the value set of $1$-forms for plane branches

TL;DR

This paper demonstrates that the value set of 1-forms uniquely determines the semigroup of a plane branch and provides an effective method to recover the semigroup from this set, aiding classification.

Contribution

It establishes that the value set of 1-forms fully determines the semigroup and introduces a method to recover the semigroup from the value set.

Findings

The value set of 1-forms determines the semigroup of a plane branch.

An effective method to recover the semigroup from the value set is provided.

It is possible to decide if a subset of natural numbers is a valid value set of 1-forms.

Abstract

The value semigroup $\Gamma$ and the value set $\Lambda$ of $1$-forms are, respectively, a topological and an analytical invariant of a plane branch. Giving a plane branch $\mathcal{C}$ with semigroup $\Gamma$ there are a finitely number of distinct possible sets $\Lambda_i$ according to the analytic class of $\mathcal{C}$. In this work we show that the value set of $1$-forms $\Lambda$ determines the semigroup $\Gamma$ and we present an effective method to recover $\Gamma$ by $\Lambda$. In particular, this allows us to decide if a subset of $\mathbb{N}$ is a value set of $1$-forms for a plane branch.