Lower bounds for the Hilbert number of polynomial systems

Let $H(m)$ denote the maximal number of limit cycles of polynomial systems of degree m. It is called the Hilbert number. The main part of Hilbert?s 16th problem posed in 1900 is to find its value. The problem is still open even for $m=2$. However, there have been many interesting results on the lower bound of it for $m?2$. In this paper, we give some new lower bounds of this number. The results obtained in this paper improve all existing results for all $m?7$ based on some known results for $m=3,4,5,6$. In particular, we obtain that $H(m)$ grows at least as rapidly as $\frac{1}{2\ln 2}{(m+2)}^2\ln (m+2)$ for all large m.