2025/04/13 by Xavier Caruso, Caruso, Xavier, Florian Fürnsinn +3 · 2 citations
Computer Science · Mathematics · #Polynomial and algebraic computation #Commutative Algebra and Its Applications #Algebraic Geometry and Number Theory
paper · pdf · doi:10.48550/arxiv.2504.09429
The aim of this paper is to investigate the algebraicity behavior of reductions of D-finite power series modulo prime numbers. For many classes of D-finite functions, such as diagonals of multivariate algebraic series or hypergeometric functions, it is known that their reductions modulo prime numbers, when defined, are algebraic. We formulate a conjecture that uniformizes the Galois groups of these reductions across different prime numbers. We then focus on hypergeometric functions, which serves as a test case for our conjecture. Refining the construction of an annihilating polynomial for the reduction of a hypergeometric function modulo a prime number p, we extract information on the respective Galois groups and show that they behave nicely as p varies.