Appears in collection : Combinatorics and Arithmetic for Physics

In 1974, Honda proved the $p$-curvature conjecture for order one differential equations with rational coefficients over a number field. He demonstrated that in this setting, the p-curvature conjecture was equivalent to a theorem due to Kronecker, providing a local-global criterion for the splitting of polynomials over the rational numbers. In 1985 the Chudnovskys published another proof of Honda’s theorem (and of Kronecker’s theorem) by means of Padé approximation and elementary number theory, thus paving the way to an effective version of these results. Here, by ”effective” we mean that we wish to obtain an explicit finite bound on the number of $p$-curvatures to be computed in order to decide the algebraicity of the solution of the differential equation. In this talk, I will explain how to obtain such a bound, and report on an implementation. This is joint work with Florian Fürnsinn (University of Vienna).