Conjectures on stably Newton degenerate singularities

TL;DR

This paper investigates whether all functions can be made non-degenerate via stabilization, concluding that some singularities remain stably degenerate, especially in finite characteristic, challenging a conjecture by Arnold.

Contribution

It introduces a method to attempt non-degeneracy through stabilization and provides examples where this method fails, supporting the conjecture of stable degeneracy for certain singularities.

Findings

Some singularities cannot be made non-degenerate after stabilization.

Finite characteristic singularities like $x^p+x^q$ are stably degenerate.

Irreducible plane curves with multiple Puiseux pairs are stably non-degenerate.

Abstract

We discuss a problem of Arnold, whether every function is stably equivalent to one which is non-degenerate for its Newton diagram. We argue that the answer is negative. We describe a method to make functions non-degenerate after stabilisation and give examples of singularities where this method does not work. We conjecture that they are in fact stably degenerate, that is not stably equivalent to non-degenerate functions. We review the various non-degeneracy concepts in the literature. For finite characteristic we conjecture that there are nowild vanishing cycles for non-degenerate singularities. This implies that the simplest example of singularities with finite Milnor number, $x^p+x^q$ in characteristic $p$, is not stably equivalent to a non-degenerate function. We argue that irreducible plane curves with an arbitrary number of Puiseux pairs (in characteristic zero) are stably non-degenerate. As the stabilisation involves many variables, it becomes very difficult to determine the Newton diagram in general, but the form of the equations indicates that the defining functions are non-degenerate.