Experimental Results for the Poincaré Center Problem
by
Hans-Christian Graf v. Bothmer
Universität Hannover
Coauthors: Martin Cremer

In 1885 Poincaré asked when the differential equation

. y   = - x + p(x, y) y+q(x, y)

has stable solutions in the neighbourhood of the equilibrium solution (x, y)=(0, 0). He showed that for p and q polynomials of fixed degree the variety X of such differential equations is algebraic, by giving an infinite generating system of polynomial equations.

For p and q homogeneous of degree 2 the variety X has been described geometrically by various autors (Dulac 1908, Frommer 1933, ..., Schlomiuk 1993). In this talk these results are reviewed and some new results for the case p and q inhomogenous of degree 3 are obtained. The methods used are reduction to finite characteristic, computer algebra and syzygies.