Abstract

We study a condition of non-degeneracy on the class of germs of analytic vector fields $X$ in a neighborhood of the origin of $Rn$ that is inspired on the Newton non-degeneracy condition of Kouchnirenko and Ehlers.
This condition leads to a result on the invariance of the local index of $X$, whenever $X-1(0)=\; 0$, under the addition of some specific monomials to each component function of the vector field. The result thus obtained generalizes the previous result of Cima-Gasull-Torregrosa on the index of semi-weighted-homogeneous vector fields and the result of Gutierrez-Rums on the index of planar vector fields and the Newton non-degeneracy condition. Our approach is based on the estimation of Lojasiewicz exponents and the computation of multiplicities through Newton polyhedra.
We also show an indication of how to give an upper bound, using the program "Singular", for the minimum of those integers $r$ such that $jrX-1(0)=\; 0$ and $ix\_0(jrX)=\; ix$₀(X), where $X$ is an analytic vector field whose complexification has an isolated singularity at the origin, the map $jrX$ is the $r$-jet of $X$ and $ix$₀(X) denotes the local index of $X$