Whitney regularity and Thom condition for families of non-isolated mixed singularities

TL;DR

This paper studies conditions under which families of mixed polynomial functions with non-isolated singularities maintain consistent geometric and topological properties, using Newton polygon techniques to establish Whitney and Thom equisingularity.

Contribution

It introduces elementary Newton polygon-based conditions that ensure Whitney and Thom equisingularity for families of mixed hypersurfaces with non-isolated singularities.

Findings

Families satisfying the conditions are Whitney equisingular.

Such families also satisfy the Thom condition.

The approach simplifies checking equisingularity via Newton polygons.

Abstract

We investigate the equisingularity question for $1$-parameter deformation families of mixed polynomial functions $f_t(\mathbf{z},\bar{\mathbf{z}})$ from the Newton polygon point of view. We show that if the members $f_t$ of the family satisfy a number of elementary conditions, which can be easily described in terms of the Newton polygon, then the corresponding family of mixed hypersurfaces $f_t^{-1}(0)$ is Whitney equisingular (and hence topologically equisingular) and satisfies the Thom condition.