Bernstein polynomial and value set of differentials for plane branches

TL;DR

This paper explores the relationship between Bernstein polynomial roots and the value set of differentials for plane branches, providing descriptions for semiquasihomogeneous cases.

Contribution

It establishes a connection between Bernstein polynomial roots and differential value sets for plane branches, especially for semiquasihomogeneous polynomials.

Findings

Identifies common roots of Bernstein polynomial for fixed differential value sets

Describes the relation for semiquasihomogeneous polynomial plane branches

Provides a framework linking Bernstein roots and differential value sets

Abstract

In this work, we study the relation between the roots of the Bernstein polynomial $\widetilde{b}$ and the value set of differentials $\Lambda$ for plane branches. For plane branches defined by semiquasihomogeneous polynomial we describe the set of common roots of $\widetilde{b}$ sharing by every branch with a fixed $\Lambda$ set.