Experimental results for the Poincar\'e center problem (including an Appendix with Martin Cremer)

TL;DR

This paper employs a heuristic finite field counting method to analyze the Poincaré center problem, confirming its accuracy for certain cases and providing new evidence supporting a conjecture about degree 3 nonlinearities.

Contribution

It introduces a heuristic approach based on finite field point counting for the Poincaré center problem, validating it for specific degrees and offering evidence for an existing conjecture.

Findings

Method correctly predicts results for degree 2 and 3 nonlinearities

Provides new evidence supporting Zoladek's conjecture for degree 3

Demonstrates effectiveness of finite field techniques in dynamical systems

Abstract

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