Toric resolutions of strongly mixed weighted homogeneous polynomial germs of type $J_{10}^-$

Abstract

We consider toric resolutions of some strongly mixed weighted homogeneous polynomials of type $J_{10}^-$. We show that the strongly mixed weighted homogeneous polynomial $f := f_{2,2,1,2,1,4}\ (k=3)$ (see \S 3) has no mixed critical points on ${\mathbb{C}^*}^2$ (Lemma 14), and moreover, show that the strict transform $\tilde V$ of the mixed hypersurface singularity $V := f^{-1}(0)$ via the toric modification $\hat{\pi} : X \to \mathbb{C}^2$, where we set $f := f_{2,2,1,2,1,4}\ (k=3)$, is not only a real analytic manifold outside of ${\tilde V} \cap \hat{\pi}^{-1}(\boldsymbol{0})$ but also a real analytic manifold as a germ of ${\tilde V} \cap \hat{\pi}^{-1}(\boldsymbol{0})$ (Theorem 15).