Experimental Results for the Poincaré

We apply a heuristic method based on counting points over finite fields to the Poincaré center problem. We show that this method gives the correct results for homogeneous non linearities of degree 2 and 3. Also we obtain some new information about general degree 3 non linearities. Introduction In 1885 Poincaré asked when the differential equation y = − x+ p(x, y) y + q(x, y) =: − P (x, y) Q(x, y with convergent power series p(x, y) and q(x, y) starting with quadratic terms, has stable solutions in the neighborhood of the equilibrium solution (x, y) = (0, 0). This means that in such a neighborhood the solutions of the equivalent plane autonomous system ẋ = y + q(x, y) = Q(x, y) ẏ = −x− p(x, y) = −P (x, y) are closed curves around (0, 0). Poincaré showed that one can iteratively find a formal power series F = x2 + y2 + f2(x, y) + f3(x, y) + . . . such that det (