Mixed functions of strongly polar weighted homogeneous face type

TL;DR

This paper studies the topology of singularities of mixed polynomials with strongly non-degenerate face functions, using toric and polar modifications to describe the Milnor fibration's zeta function.

Contribution

It introduces a method to resolve singularities of mixed polynomials and provides a Varchenko-type formula for the zeta function of their Milnor fibrations.

Findings

Toric modification resolves the singularity topologically.

The zeta function of the Milnor fibration is explicitly described.

The approach applies to mixed polynomials with strongly non-degenerate face functions.

Abstract

Let $f(\bf z,\bar{\bf z})$ be a mixed polynomial with strongly non-degenerate face functions. We consider a canonical toric modification $\pi:\,X\to \Bbb C^n$ and a polar modification $\pi_{\Bbb R}:Y\to X$. We will show that the toric modification resolves topologically the singularity of $V$ and the zeta function of the Milnor fibration of $f$ is described by a formula of a Varchenko type.