Resolutions of Newton non-degenerate mixed polynomials of strongly polar non-negative mixed weighted homogeneous face type

TL;DR

This paper demonstrates that a specific toric modification can topologically resolve singularities of certain mixed hypersurfaces defined by Newton non-degenerate mixed polynomials with strongly polar non-negative mixed weighted homogeneous face functions.

Contribution

It extends previous results by showing topological resolution of singularities for a broader class of mixed polynomials using toric modifications.

Findings

Toric modification resolves singularities under certain conditions.

Extension of Oka's theorem to mixed polynomial case.

Examples illustrating the resolution process.

Abstract

Let $f(\mathbb{z},\bar{\mathbb{z}})$ be a convenient Newton non-degenerate mixed polynomial with strongly polar non-negative mixed weighted homogeneous face functions. We consider a convenient regular simplicial cone subdivision $\Sigma^*$ which is admissible for $f$ and take the toric modification $\hat{\pi} : X \to \mathbb{C}^n$ associated with $\Sigma^*$. We show that the toric modification resolves topologically the singularity of the mixed hypersurface germ defined by $f(\mathbb{z},\bar{\mathbb{z}})$ under the Assumption (*) (Theorem 32). This result is an extension of the first part of Theorem 11 ([4]) by Mutsuo Oka. We also consider some typical examples (\S 9).