An Effective Proof of the p-curvature Conjecture for First-order Differential Equations

Explore an advanced mathematical lecture examining an effective proof of the p-curvature conjecture for first-order differential equations with rational coefficients. Delve into Honda's 1974 breakthrough that established the equivalence between the p-curvature conjecture and Kronecker's theorem for order one differential equations over number fields, providing crucial local-global criteria for polynomial splitting over rational numbers. Learn about the Chudnovskys' 1985 alternative proof using Padé approximation and elementary number theory, which opened pathways to effective versions of these fundamental results. Understand the concept of "effective" proofs in this context, focusing on obtaining explicit finite bounds for the number of p-curvatures required to determine the algebraicity of differential equation solutions. Discover the speaker's collaborative work with Florian Fürnsinn from the University of Vienna in developing computational implementations and explicit bounds for this important mathematical problem. Gain insights into the intersection of differential equations, algebraic number theory, and computational mathematics through this detailed exposition of cutting-edge research in the field.