Generalized upper principal part of real planar polynomial vector fields

Abstract

The goal of this paper is to generalize recent results about the topological classification of the dynamics of a real planar polynomial vector field near infinity. Given a polynomial vector field $X$, using Newton polyhedra one can define its generalized upper principal part $X_{\Gamma }^U$. By dropping some monomials of $X_{\Gamma }^U$, we define its minimal generalized upper principal part $X_G^U$. We prove that there exist an open and dense set ${\tilde{\mathfrak{U}}}_1$ in the set of polynomial vector fields with Newton degenerate upper principal part satisfying the following property: if $X\in {\tilde{\mathfrak{U}}}_1$, then $X$ and $X_G^U$ are topologically equivalent near infinity, provided that $X_G^U$ satisfies some additional non-degeneracy assumptions. Our techniques rely on toric compactification and the Normal Form Theorem.

30 pages, 8 figures