On the Milnor fibration of certain Newton degenerate functions

TL;DR

This paper extends the understanding of Milnor fibrations by showing that, for certain degenerate polynomial functions, their diffeomorphism type is determined by the Newton boundaries of their factors, under specific non-degeneracy conditions.

Contribution

It generalizes the known result for non-degenerate functions to a class of degenerate functions formed by products, under non-degenerate complete intersection conditions.

Findings

Diffeomorphism type determined by Newton boundaries for certain degenerate functions.

Extension of Milnor fibration classification to degenerate polynomial functions.

Conditions involving non-degenerate complete intersections are crucial.

Abstract

It is well known that the diffeomorphism-type of the Milnor fibration of a (Newton) non-degenerate polynomial function $f$ is uniquely determined by the Newton boundary of $f$. In the present paper, we generalize this result to certain degenerate functions, namely we show that the diffeomorphism-type of the Milnor fibration of a (possibly degenerate) polynomial function of the form $f=f^1\cdots f^{k_0}$ is uniquely determined by the Newton boundaries of $f^1,\ldots, f^{k_0}$ if $\{f^{k_1}=\cdots=f^{k_m}=0\}$ is a non-degenerate complete intersection variety for any $k_1,\ldots,k_m\in \{1,\ldots, k_0\}$.