Inner and Partial non-degeneracy of mixed functions

TL;DR

This paper extends the concept of non-degeneracy to mixed polynomials in complex variables and their conjugates, analyzing singularities and Milnor fibrations.

Contribution

It introduces new notions of partial non-degeneracy for mixed functions and compares them with existing non-degeneracy types, establishing their implications for singularities.

Findings

Partial non-degeneracy implies weakly isolated singularities.

Strong partial non-degeneracy implies isolated singularities.

Strong inner non-degeneracy ensures the strong Milnor condition.

Abstract

Mixed polynomials $f:\mathbb{C}^2\to\mathbb{C}$ are polynomial maps in complex variables $u$ and $v$ as well as their complex conjugates $\bar{u}$ and $\bar{v}$. They are therefore identical to the set of real polynomial maps from $\mathbb{R}^4$ to $\mathbb{R}^2$. We generalize Mondal's notion of partial non-degeneracy from holomorphic polynomials to mixed polynomials, introducing the concepts of partially non-degenerate and strongly partially non-degenerate mixed functions. We prove that partial non-degeneracy implies the existence of a weakly isolated singularity, while strong partial non-degeneracy implies an isolated singularity. We also compare (strong) partial non-degeneracy with other types of non-degeneracy of mixed functions, such as (strong) inner non-degeneracy, and find that, in contrast to the holomorphic setting, the different properties are not equivalent for mixed polynomials. We then introduce additional conditions under which strong partial non-degeneracy becomes equivalent to the existence of an isolated singularity. Furthermore, we prove that mixed polynomials that are strongly inner non-degenerate satisfy the strong Milnor condition, resulting in an explicit Milnor (sphere) fibration.