Conjectures on Stably Newton Degenerate Singularities
Artikel i vetenskaplig tidskrift, 2021

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 no wild vanishing cycles for non-degenerate singularities. This implies that the simplest example of singularities with finite Milnor number, xp+ xq 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.

Newton diagram

Wild vanishing cycles


Stable equivalence


Jan Stevens

Chalmers, Matematiska vetenskaper, Algebra och geometri

Göteborgs universitet

Arnold Mathematical Journal

21996792 (ISSN) 21996806 (eISSN)

Vol. 7 3 441-465


Algebra och logik


Matematisk analys



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