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arXiv · 1711.04093

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

Guangfeng Dong·Changjian Liu·Jiazhong Yang
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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 $2 n^2$.

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Guangfeng Dong, Changjian Liu, Jiazhong Yang. 2017-11-11. On the maximal saddle order of p:-q resonant saddle. https://arxiv.org/abs/1711.04093

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