On the Milnor fibration for $f(z)\bar g(z)$ II

TL;DR

This paper investigates the Milnor fibration properties of mixed functions of the form $f(z)ar g(z)$, establishing conditions under which they admit tubular and spherical Milnor fibrations and proving their equivalence.

Contribution

It extends the understanding of Milnor fibrations to mixed functions without the convenience assumption, demonstrating their existence and equivalence under non-degeneracy and tameness conditions.

Findings

Existence of tubular Milnor fibration for $H(z,ar z)=f(z)ar g(z)$

Existence of spherical Milnor fibration for the same class of functions

Equivalence between the tubular and spherical Milnor fibrations

Abstract

We consider a mixed function of type $H(z,\bar z)=f(z)\bar g(z)$ where $f,g$ are non-degenerate but they are not assumed to be convenient. We assume that $f=0$ and $g=0$ and $f=g=0$ are non-degenerate and locally tame. We will show that $H$ has a tubular Milnor fibration and a spherical Milnor fibration. We show also two fibrations are equivalent.