Geometric Triviality of the Strongly Minimal Second Painlev\'e equations

TL;DR

This paper proves that the strongly minimal second Painlevé equation exhibits geometric triviality, meaning algebraic dependencies among solutions imply dependencies among pairs, extending previous results to non-generic parameters.

Contribution

It extends the geometric triviality result of the second Painlevé equation to non-generic parameters, clarifying the algebraic dependence structure of its solutions.

Findings

Algebraic dependencies among multiple solutions imply pairwise dependencies.

The result applies to non-generic parameter values.

Extends recent findings on the second Painlevé equation.

Abstract

We show that the strongly minimal second Painlev\'e equation (y" = 2y^3+ty+\alpha) 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 Painlev\'e equation to the non generic parameters.