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arXiv · 2312.00217

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

Thais Maria Dalbelo·Regilene Oliveira·Otavio Henrique Perez
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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.

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Thais Maria Dalbelo, Regilene Oliveira, Otavio Henrique Perez. 2023-11-30. Topological equivalence in the infinity of a planar vector field and its principal part defined through Newton polytope. https://arxiv.org/abs/2312.00217

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