On some generalizations of Newton non degeneracy for hypersurface singularities

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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.