The index of analytic vector fields and Newton polyhedra

Carles Bivià-Ausina¹

¹ Departament de Matemàtica Aplicada Universitat Politècnica de València Escola Politècnica Superior d'Alcoi Plaça Ferrándiz i Carbonell 2 03801 Alcoi (Alacant), Spain

Fundamenta Mathematicae, Tome 177 (2003) no. 3, pp. 251-267.

Résumé

We prove that if $f:(\mathbb R^n,0)\to (\mathbb R^n,0)$ is an analytic map germ such that $f^{-1}(0)=\{0\}$ and $f$ satisfies a certain non-degeneracy condition with respect to a Newton polyhedron ${\mit\Gamma}_+\subseteq\mathbb R^n$, then the index of $f$ only depends on the principal parts of $f$ with respect to the compact faces of ${\mit\Gamma}_+$. In particular, we obtain a known result on the index of semi-weighted-homogeneous map germs. We also discuss non-degenerate vector fields in the sense of Khovanski\u\i and special applications of our results to planar analytic vector fields.

Détail

DOI : 10.4064/fm177-3-5

Keywords: prove mathbb mathbb analytic map germ satisfies certain non degeneracy condition respect newton polyhedron mit gamma subseteq mathbb index only depends principal parts respect compact faces mit gamma particular obtain known result index semi weighted homogeneous map germs discuss non degenerate vector fields sense khovanski special applications results planar analytic vector fields