UNCW Department of Mathematics and Statistics Systems”

UNCW Department of Mathematics and
“Lower Bounds for the Hilbert Number of Polynomial
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
(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