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

Abstract

A mixed polynomial $f(\mathbf{z},\bar{\mathbf{z}})$ 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 $f:{{\mathbb{C}}^{\ast }}^n\to \mathbb{C}$ has no mixed critical points. Moreover, under the assumption $f^{-1}(0)\cap {{\mathbb{C}}^{\ast }}^n\ne \varnothing$, $f:{{\mathbb{C}}^{\ast }}^n\to \mathbb{C}$ 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 $f^{-1}(0)\cap {{\mathbb{C}}^{\ast }}^n$, and show the existence of a collection of mixed weighted homogeneous polynomials $f=f_{\Delta (P)}$ of non-zero polar degree which satisfy $\dim \Delta (P)\ge 1$ and $f^{-1}(0)\cap {{\mathbb{C}}^{\ast }}^n=\varnothing$ (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).

11 pages

References in corpus (2)

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