Geometric Triviality of the Strongly Minimal Second Painlevé equations
arXiv:1302.4338 · doi:10.1016/j.apal.2014.11.008
Abstract
We show that the strongly minimal second Painlevé equation (y" = 2y^3+ty+α) is geometrically trivial, that is we show that if y_1,...,y_n are distinct solutions such that y_1,y_1',y_2,y_2',...,y_n,y_n' are algebraically dependent over C(t), then already for some i<j, y_i,y_i',y_j,y_j' are algebraically dependent over C(t). This gives an extension of some recent result for the second Painlevé equation to the non generic parameters.