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

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-Padé 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.