On some generalizations of Newton non-degeneracy for hypersurface singularities

• Autores: Dmitry Kerner
• Localización: Journal of the London Mathematical Society, ISSN 0024-6107, Vol. 82, Nº 1, 2010, págs. 49-73
• Idioma: inglés
• DOI: 10.1112/jlms/jdq011
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• Resumen

  • We introduce two generalizations of Newton non-degenerate singularities of hypersurfaces.

    Roughly speaking, an isolated hypersurface singularity is called topologically Newton nondegenerate 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