A note on Newton non-degeneracy of mixed weighted homogeneous polynomials

Abstract

A mixed polynomial $f(\boldsymbol{z}, \bar{\boldsymbol{z}})$ is called a mixed weighted homogeneous polynomial (Definition 5) if it is both radially and polar weighted homogeneous. Let $f$ be a mixed weighted homogeneous polynomial with respect to a strictly positive radial weight vector $P$ and a polar weight vector $Q$. Suppose that $f$ is Newton non-degenerate over a compact face $Δ(P)$ and polar weighted homogeneous of non-zero polar degree with respect to $Q$. Then $f : {\mathbb{C}^*}^n \to \mathbb{C}$ has no mixed critical points. Moreover, under the assumption $f^{-1}(0) \cap {\mathbb{C}^*}^n \neq \emptyset$, $f : {\mathbb{C}^*}^n \to \mathbb{C}$ is surjective. In other words, in this situation, Newton non-degeneracy over a compact face $Δ(P)$ implies strong Newton non-degeneracy over $Δ(P)$ (Proposition 10). With this fact as a starting point, we investigate the sets $f^{-1}(0) \cap {\mathbb{C}^*}^n$, and show the existence of a collection of mixed weighted homogeneous polynomials $f = f_{Δ(P)}$ of non-zero polar degree which satisfy $\dim Δ(P) \geq 1$ and $f^{-1}(0) \cap {\mathbb{C}^*}^n = \emptyset$ (Theorem 11). We also give an example of convenient mixed function germs of mixed weighted homogeneous face type which are not true non-degenerate (Definition 14).