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UNCW Department of Mathematics and Statistics “Lower Bounds for the Hilbert Number of Polynomial Systems” Professor Maoan Han Department of Mathematics Shanghai Normal University Abstract: 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 1902 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 talk, we give some new lower bounds of this number. The results obtained improve all existing results for all m ≥ 7 based on some known results for m = 3,4,5,6. . In particular, we confirm the conjecture H (2k ) + 1 ≥ (2k + 1) 2 − 1 and obtain that H (m) grows at least as rapidly as 1 (m + 2) 2 ln(m + 2) 2 ln 2 for all large H (m) . Time: November 03, 2008, 2:00 pm – 3:30 pm, Teatime: 3:30 pm – 4:00 pm Location: 219 Bear Hall