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. In Press


Algebra och logik


Matematisk analys



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