On the maximal saddle order of p:-q resonant saddle

TL;DR

This paper investigates the growth of the maximal saddle order in non-integrable resonant saddles of planar polynomial vector fields, establishing lower bounds that grow quadratically with the degree of the system.

Contribution

It provides new estimations for the saddle order growth and identifies bounds for saddle values not generated by earlier values in resonant saddle systems.

Findings

Maximal saddle order can grow at least as rapidly as n^2.

Existence of an integer k_0 with specific properties related to saddle values.

Sharper bounds for p=1 or q=1 cases, growing as at least 2n^2.

Abstract

In this paper, we obtain some estimations of the saddle order which is the sole topological invariant of the non-integrable resonant saddles of planar polynomial vector fields of arbitrary degree $n$. Firstly, we prove that, for any given resonance $p:-q$, $(p, q)=1$, and sufficiently big integer $n$, the maximal saddle order can grow at least as rapidly as $n^2$. Secondly, we show that there exists an integer $k_0$, which grows at least as rapidly as $3n^2/2$, such that $L_{k_0}$ does not belong to the ideal generated by the first $k_0-1$ saddle values $L_1, L_2, \cdots, L_{k_0-1}$, where $L_{k}$ means the $k$-th saddle value of the given system. In particular, if $p=1$ (or $q=1$), we obtain a sharper result that $k_0$ can grow at least as rapidly as $2 n^2$.