Joan Torregrosa (UAB, Barcelona, Spain)

Wednesday, May 9th, 16:20, Room 3-010

**Title: New lower bounds for the Hilbert numbers using reversible centers**

**Abstract:** In this talk we provide the best lower bounds, that are known up to now, for the Hilbert numbers of polynomial vector fields of degree $N$, $H(N)$, for small values of $N$. These limit cycles appear bifurcating from new symmetric Darboux reversible centers with very high simultaneous cyclicity. The considered systems have, at least, three centers, one on the reversibility straight line and two symmetric about it. More concretely, the limit cycles are in a three nests configuration and the total number of limit cycles is at least $2n+m$, for some values of $n$ and $m$. The new lower bounds are obtained using simultaneous degenerate Hopf bifurcations. In particular, $H(4)\ge 28,$ $H(5)\ge 37,$ $H(6)\ge 53,$ $H(7)\ge 74,$ $H(8)\ge 96,$ $H(9)\ge 120,$ and $H(10)\ge 142.$