Links of singularities of inner non-degenerate mixed functions

TL;DR

This paper introduces the concept of inner non-degenerate mixed functions, characterizes their singularities and links via Newton boundary, and constructs infinite families of real algebraic links in the 3-sphere.

Contribution

It defines inner non-degenerate mixed functions, relates their singularities to Newton boundary, and characterizes their links, enabling the construction of infinite real algebraic links.

Findings

Inner non-degenerate mixed polynomials have weakly isolated singularities.

Strongly inner non-degenerate mixed polynomials have isolated singularities.

Links of these singularities can be characterized by the Newton boundary.

Abstract

We introduce the notion of a (strongly) inner non-degenerate mixed function $f:\mathbb{C}^2\to\mathbb{C}$. We show that inner non-degenerate mixed polynomials have weakly isolated singularities and strongly inner non-degenerate mixed polynomials have isolated singularities. Furthermore, under one additional assumption, which we call "niceness", the links of these singularities can be completely characterized in terms of the Newton boundary of $f$. In particular, adding terms above the Newton boundary does not affect the topology of the link. This allows us to define an infinite family of real algebraic links in the 3-sphere.