发表机构
Federal University of São Carlos (UFSCar); University of São Paulo (USP), Institute of Mathematics and Computer Science（圣卡洛斯联邦大学; 圣保罗大学数学与计算机科学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文推广实平面多项式向量场无穷远动力学拓扑分类的最新结果，利用牛顿多面体定义广义上主部与极小广义上主部，证明在特定开稠密集里满足非退化假设时原向量场与极小广义上主部无穷远拓扑等价，依托环面紧化与标准型定理。
AI 中文摘要
本文旨在推广关于实平面多项式向量场在无穷远处动力学拓扑分类的最新结果。给定多项式向量场$X$，利用牛顿多面体可定义其广义上主部$X_{\boldsymbol{\textit{\textbackslash Gamma}}}^{U}$；通过舍弃$X_{\boldsymbol{\textit{\textbackslash Gamma}}}^{U}$的部分单项式，可定义其极小广义上主部$X_{G}^{U}$。我们证明，在具有牛顿退化上主部的多项式向量场集合中，存在一个开且稠密的集合$\boldsymbol{\textit{\textbackslash widetilde{\textbackslash mathfrak{U}}}}_{1}$，满足：若$X$属于$\boldsymbol{\textit{\textbackslash widetilde{\textbackslash mathfrak{U}}}}_{1}$，且$X_{G}^{U}$满足若干额外非退化假设，则$X$与$X_{G}^{U}$在无穷远处拓扑等价。本文的技术方法依托于环面紧化与标准型定理。
英文摘要
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 $X_Γ^{U}$. By dropping some monomials of $X_Γ^{U}$, we define its minimal generalized upper principal part $X_{G}^{U}$. We prove that there exist an open and dense set $\widetilde{\mathfrak{U}}_{1}$ in the set of polynomial vector fields with Newton degenerate upper principal part satisfying the following property: if $X\in\widetilde{\mathfrak{U}}_{1}$, 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.
Comments30 pages, 8 figures
