Fold singularities of the maps associated with Milnor fibration for mixed polynomials

TL;DR

This paper investigates the singularities of Milnor fibration product maps for mixed polynomials, providing a characterization of singular points and criteria for fold singularities in specific cases, advancing understanding of their topological structure.

Contribution

It introduces a new characterization of singular points and a fold singularity criterion for Milnor fibration product maps associated with mixed polynomials.

Findings

Characterization of singular points of Milnor fibration product maps.

Criterion for fold singularities in polar weighted homogeneous polynomials.

Extension of Milnor fibration analysis to mixed polynomials.

Abstract

Milnor fibrations were extended by Mutsuo Oka for certain mixed polynomial. In this paper, we study singular points of differentiable maps into the 2-dimensional torus, called Milnor fibration product maps, obtained by several Milnor fibrations for mixed polynomial. We give a characterization of singular points of such product maps, and for the case of certain polar weighted homogeneous polynomials, a criterion for a fold singular point.