Experimental results for the Poincar\'e center problem (including an   Appendix with Martin Cremer)
  We apply a heuristic method based on counting points over finite fields to the Poincar\'e center problem. We show that this method gives the correct results for homogeneous non linearities of degree 2 and 3. Also we obtain new evidence for Zoladek's conjecture about general degree 3 non linearities 
Hans-Christian Graf v. Bothmer