On Milnor fibrations of mixed functions, $a_f$-condition and boundary stability

TL;DR

This paper extends the study of Milnor fibrations to non-convenient mixed functions, analyzing geometric conditions like Thom's $a_f$ condition and boundary stability to understand their singularity structures.

Contribution

It introduces new insights into Milnor fibrations of non-convenient mixed functions and explores their geometric properties and stability conditions.

Findings

Milnor fibrations exist for certain non-convenient mixed functions.

Thom's $a_f$ condition relates to boundary stability in these fibrations.

Transversality of nearby fibers is characterized in this context.

Abstract

Convenient mixed functions with strongly non-degenerate Newton boundaries have Milnor fibrations, as the isolatedness of the singularity follows from the convenience. In this paper, we consider the Milnor fibration for non-convenient mixed functions. We also study geometric properties such as Thom's $a_f$ condition, the transversality of the nearby fibers and stable boundary property of the Milnor fibration and their relations.