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

Abstract

Let $f(\mathbf z,\bar{\mathbf z})$ 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 $f(\mathbf z,\bar{\mathbf z})$ under the Assumption(*) (Theorem 32). This result is an extension of the first part of Theorem 11 ([4]) by M. Oka, which studies strongly polar positive cases, to strongly polar non-negative cases. We also consider some typical examples (§9).