Topological equivalence in the infinity of a planar vector field and its principal part defined through Newton polytope

TL;DR

This paper shows that for planar polynomial vector fields, the phase portrait near infinity is determined by the monomials on the upper boundary of the Newton polytope, under certain conditions, linking algebraic structure to topological dynamics.

Contribution

It establishes a topological equivalence between the phase portrait near infinity and the principal part defined by the Newton polytope, extending previous results to a broader class of vector fields.

Findings

The monomials on the upper boundary of the Newton polytope determine the phase portrait near infinity.

The result generalizes classical theorems by Berezovskaya, Brunella, and Miari.

The effect of Poincaré--Lyapunov compactification on the Newton polytope is analyzed.

Abstract

Given a planar polynomial vector field $X$ with a fixed Newton polytope $\mathcal{P}$, we prove (under some non degeneracy conditions) that the monomials associated to the upper boundary of $\mathcal{P}$ determine (under topological equivalence) the phase portrait of $X$ in a neighbourhood of boundary of the Poincar\'e--Lyapunov disk. This result can be seen as a version of the well known result of Berezovskaya, Brunella and Miari for the dynamics at the infinity, We also discuss the effect of the Poincar\'e--Lyapunov compactification on the Newton polytope.