On the maximal saddle order of <inline-formula><tex-math id="M1">\begin{document}$ p:-q $\end{document}</tex-math></inline-formula> resonant saddle

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 , • • • , L k 0 −1 , where L k represents 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 2n 2. These results are valid for both cases of real and complex resonant saddles.