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.

翻译：基于Chudnovsky兄弟提出的来自Hermite-Padé逼近的渐近论证，我们发展了多项式分裂的Kronecker定理的一个有效版本。结合Honda关于具有多项式系数的一阶方程$p$曲率猜想的证明，我们利用该结果推导出Grothendieck $p$曲率猜想在一阶方程情形下的有效版本。更精确地说，我们给出了给定微分方程的$p$曲率必须为零的素数个数的上界（该上界用系数的高度和次数表示），从而判定其具有非零代数解。基于此方法，我们描述了一个利用$p$曲率判定一阶微分方程代数解存在的算法，并报告了其在SageMath中的实现。