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

TL;DR

This paper provides a new, elementary proof of Deligne's theorem, establishing a polynomial bound on the algebraic degrees of reductions modulo p of multivariate algebraic power series, with explicit degree estimates.

Contribution

It offers the first explicit polynomial degree bound on algebraic degrees modulo p, improving understanding of algebraic power series reductions.

Findings

Established a polynomial bound on algebraic degrees d_p

Provided an explicit degree estimate for these bounds

Simplified the proof of Deligne's theorem

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.