We prove that if f:(ℝⁿ,0) → (ℝⁿ,0) is an analytic map germ such that and f satisfies a certain non-degeneracy condition with respect to a Newton polyhedron Γ₊ ⊆ ℝⁿ, then the index of f only depends on the principal parts of f with respect to the compact faces of Γ₊. 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 Khovanskiĭand special applications of our results to planar analytic vector fields.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm177-3-5,
author = {Carles Bivi\`a-Ausina},
title = {The index of analytic vector fields and Newton polyhedra},
journal = {Fundamenta Mathematicae},
volume = {177},
year = {2003},
pages = {251-267},
zbl = {1057.32011},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm177-3-5}
}
Carles Bivià-Ausina. The index of analytic vector fields and Newton polyhedra. Fundamenta Mathematicae, Tome 177 (2003) pp. 251-267. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm177-3-5/