An Effective Version of the $p$-Curvature Conjecture for Order One Differential Equations

Abstract: We develop an effective version of Kronecker's Theorem on the splitting of polynomials, based on asymptotic arguments proposed by the Chudnovsky brothers, coming from Hermite-Pad\'e approximation. In conjunction with Honda's proof of the $p$-curvature conjecture for order one equations with polynomial coefficients we use this to deduce an effective version of the Grothendieck $p$-curvature conjecture for order one equations. More precisely, we bound the number of primes for which the $p$-curvature of a given differential equation has to vanish in terms of the height and the degree of the coefficients, in order to conclude it has a non-zero algebraic solution. Using this approach, we describe an algorithm that decides algebraicity of solutions of differential equation of order one using $p$-curvatures, and report on an implementation in SageMath.