Uniform stable radius and Milnor number for non-degenerate isolated complete intersection singularities

TL;DR

This paper proves that for certain non-degenerate isolated complete intersection singularities, the Milnor number remains invariant under specific conditions, and provides a formula for it based on Newton polyhedra.

Contribution

It introduces a piecewise analytic family with a uniform stable radius for Milnor fibrations and generalizes the invariance of Milnor numbers using Newton polyhedra.

Findings

Milnor numbers are equal for non-degenerate complete intersections with same Newton polyhedra.

Provides a formula for Milnor number in terms of Newton polyhedra.

Establishes invariance of Milnor number for non-degenerate isolated singularities.

Abstract

We prove that for two germs of analytic mappings $f,g\colon (\mathbb{C}^n,0) \rightarrow (\mathbb{C}^p,0)$ with the same Newton polyhedra which are (Khovanskii) non-degenerate and their zero sets are complete intersections with isolated singularity at the origin, there is a piecewise analytic family $\{f_t\}$ of analytic maps with $f_0=f, f_1=g$ which has a so-called {\it uniform stable radius for the Milnor fibration}. As a corollary, we show that their Milnor numbers are equal. Also, a formula for the Milnor number is given in terms of the Newton polyhedra of the component functions. This is a generalization of the result by C. Bivia-Ausina. Consequently, we obtain that the Milnor number of a non-degenerate isolated complete intersection singularity is an invariance of Newton boundaries.