Initial Newton polynomial of the discriminant

TL;DR

This paper demonstrates that the initial Newton polynomial of the discriminant of a holomorphic map with an isolated zero is uniquely determined, up to rescaling, by the ideals generated by its component functions.

Contribution

It establishes a direct link between the initial Newton polynomial of the discriminant and the ideals of the component functions, providing a new understanding of their relationship.

Findings

Initial Newton polynomial is determined by the ideals $(f)$ and $(g)$.

The result holds up to rescaling of variables.

Provides a method to compute the discriminant's Newton polynomial from ideals.

Abstract

Let $(f,g)\colon (\mathbb{C}^2,0)\longrightarrow (\mathbb{C}^2,0)$ be a holomorphic mapping with an isolated zero. We show that the initial Newton polynomial of its discriminant is determined, up to rescalling variables, by the ideals $(f)$ and $(g)$.