On some generalizations of Newton non-degeneracy for hypersurface singularities

We introduce two generalizations of Newton non-degenerate singularities of hypersurfaces. Roughly speaking, an isolated hypersurface singularity is called topologically Newton non-degenerate 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 topologically Newton non-degenerate is called essentially Newton degenerate. For plane curves we give an explicit characterization of topologically Newton non-degenerate singularities; for hypersurfaces we provide several examples.Next, we treat the question of whether Newton non-degenerate or topologically Newton non-degenerate is a property of singularity types or of particular representatives: namely, is the non-degeneracy preserved in an equisingular family? This result is proved for curves. For hypersurfaces we give an example of a Newton non-degenerate hypersurface whose equisingular deformation consists of essentially Newton degenerate hypersurfaces.Finally, we define the directionally Newton non-degenerate germs, a subclass of topologically Newton non-degenerate ones. For such singularities the classical formulas for the Milnor number and the zeta function of the Newton non-degenerate hypersurface are generalized.