Diagonals and algebraicity modulo $p$: a sharper degree bound

Abstract

In 1984, Deligne proved that for any prime number $p$, the reduction modulo $p$ of the diagonal of a multivariate algebraic power series with integer coefficients is algebraic over the field of rational functions with coefficients in ${\mathbb{F}}_p$. Moreover, he conjectured that the algebraic degrees $d_p$ of these functions should grow at most polynomially in $p$. In this article, we provide a new and elementary proof of Deligne's theorem, which yields the first general polynomial bound on $d_p$ with an explicit and reasonable degree.

To appear in the Annales scientifiques de l'{É}cole normale sup{é}rieure. A longer version of this work is available at arXiv:2306.02640