Topology of the Milnor fibrations of polar weighted homogeneous polynomials

TL;DR

This paper investigates the topology of Milnor fibrations for polar weighted homogeneous polynomials, providing a concrete decomposition approach and formulas for characteristic polynomials related to singularities and their deformations.

Contribution

It introduces a round handle decomposition method for Milnor fibrations of deformed polar weighted homogeneous polynomials, linking handle counts to singularity link components.

Findings

Explicit round handle decomposition of Milnor fibrations.

Formulas for characteristic polynomials of singularities.

Relation between handle counts and singularity link components.

Abstract

In the present paper, we study deformations of polar weighted homogeneous polynomials which are also polar weighted homogeneous polynomials. We describe a round handle decomposition of the Milnor fibration of a deformation of a polar weighted homogeneous polynomial concretely and give the number of round handles by the number of positive and negative components of the links of singularities appearing before and after the deformation. We also give a formula of characteristic polynomials of these singularities by using the decomposition of the monodromy of the Milnor fibration induced by a round handle decomposition.