Some comments about Hilbert 16th Problem
主 题: Some comments about Hilbert 16th Problem
报告人: Professor Robert Roussarie (University of Bourgogne, France)
时 间: 2016-06-17 15:00 - 16:00
地 点: 理科一号楼 1114(数学所活动）
The qualitative ideas of H. Poincare for differential equations are at the background of the Hilbert 16th Problem. A weak formulation of this problem is as follow : for any n bigger than or equal to 2; there exists a finite number H(n) such that any planar polynomial vector field of degree less than n has less than H(n) isolated closed orbits. I want to say why the Hilbert problem keeps all his importance today, principally for its theoretical interest and the fact it has impulsed a flow of researches in various directions. Without entering into its very long history, I want to recall some reductions of the problem, subproblems and methods which were used until now in the treatment of the question. These methods may come from algebra, as the desingularization methods, from real or complex analysis, from asymptotic theory. I shall illustrate them with some remarkable partial results which where obtained in the direction of a proof of the Hilbert 16th Problem, which remains unsolved even for n=2.