Diagonals and algebraicity modulo p: a sharper degree bound

2026/01/21 by Boris Adamczewski, Alin Bostan, Xavier Caruso · 1 voice · 2 citations
Computer Science · Mathematics · #cs.SC #math.NT

paper · pdf · doi:10.48550/arxiv.2601.14920

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.

Citations

• Galois groups of reductions modulo p of D-finite series
• A sharper multivariate Christol's theorem with applications to diagonals and Hadamard products
• Arithmetic statistics of Galois groups
• Algebraic independence of G-functions and congruences "à la Lucas"
• Algebraic independence of G-functions and congruences "à la Lucas"
• Suites algébriques, automates et substitutions

Discussions

• Boris Adamczewski (UCBL, CIRM, CNRS), Alin Bostan (PolSys), Xavier Caruso (CNRS, IMB): Diagonals and algebraicity modulo $p$: a sharper degree bound https://arxiv.org/abs/2601.14920 https://arxiv.org/ [bsky, 0 points, 0 comments]

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