Geometric Triviality of the Strongly Minimal Second Painlev\'e equations
Abstract
We show that the strongly minimal second Painlev\'e equation (y" = 2y3+ty+α) is geometrically trivial, that is we show that if y1,...,yn are distinct solutions such that y1,y1',y2,y2',...,yn,yn' are algebraically dependent over C(t), then already for some i<j, yi,yi',yj,yj' are algebraically dependent over C(t). This gives an extension of some recent result for the second Painlev\'e equation to the non generic parameters.
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