On some generalizations of Newton non degeneracy for hypersurface singularities

TL;DR

This paper introduces two new generalizations of Newton-non-degenerate hypersurface singularities, explores their properties, and examines their invariance under deformations, providing explicit characterizations and extending classical formulas.

Contribution

It defines topologically Newton-non-degenerate singularities and explores their properties, including invariance and generalizations of classical invariants.

Findings

Characterization of tNnd singularities for plane curves

Examples of hypersurfaces illustrating the new concepts

Extension of Milnor number and zeta function formulas

Abstract

We introduce two generalizations of Newton-non-degenerate (Nnd) singularities of hypersurfaces. Roughly speaking, an isolated hypersurface singularity is called topologically Newton-non-degenerate (tNnd) if the local embedded topological singularity type can be restored from a collection of Newton diagrams (for some coordinate choices). A singularity that is not tNnd is called essentially Newton-degenerate. For plane curves we give an explicit characterization of tNnd singularities; for hypersurfaces we provide several examples. Next, we treat the question: whether Nnd or tNnd is a property of singularity types or of particular representatives. Namely, is the non-degeneracy preserved in an equi-singular family? This fact is proved for curves. For hypersurfaces we give an example of a Nnd hypersurface whose equi-singular deformation consists of essentially Newton-degenerate hypersurfaces. Finally, we define the directionally Newton-non-degenerate germs, a subclass of tNnd germs. For such singularities the classical formulas for the Milnor number and the zeta function of the Nnd hypersurface are generalized.