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

TL;DR

This paper studies the properties of mixed weighted homogeneous polynomials, showing that under certain non-degeneracy conditions, these functions have no mixed critical points and are surjective, with implications for their zero sets.

Contribution

It establishes that Newton non-degeneracy over a compact face implies strong non-degeneracy and constructs examples of mixed weighted homogeneous polynomials with specific zero set properties.

Findings

No mixed critical points under non-degeneracy conditions

Surjectivity of the polynomial when zero set intersects the torus

Existence of polynomials with empty zero set despite face dimension

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 $\Delta(P)$ 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 $\Delta(P)$ implies strong Newton non-degeneracy over $\Delta(P)$ (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 $f = f_{\Delta (P)}$ of non-zero polar degree which satisfy $\dim \Delta (P) \geq 1$ 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).