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

Boris Adamczewski, Alin Bostan, Xavier Caruso

TL;DR

The paper proves a sharp, elementary validation of Deligne's modulo-$p$ diagonal theorem by recasting diagonals as residues and employing a refined multivariate Christol framework via section operators. It establishes a concrete polynomial-type bound on the $p$-adic algebraic degree $d_p$, with $d_p \, ext{bounded by}\, p^N$ where $N$ is explicitly determined from the degree $d$ and height $h$ (and, more generally, from generalized diagonals via Newton polytopes). It introduces a generalized-diagonal machinery and uses Newton-polytope arguments to bound the Frobenius-orbit dimensions, avoiding inductive geometric arguments. The results yield effective, uniform bounds and have algorithmic consequences for computing diagonals modulo $p$, with potential connections to geometric invariants like genus. Overall, the work sharpens the understanding of how diagonals of multivariate algebraic series behave modulo primes and provides practical, polynomial bounds on their algebraic degrees.

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.

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

TL;DR

The paper proves a sharp, elementary validation of Deligne's modulo- diagonal theorem by recasting diagonals as residues and employing a refined multivariate Christol framework via section operators. It establishes a concrete polynomial-type bound on the -adic algebraic degree , with where is explicitly determined from the degree and height (and, more generally, from generalized diagonals via Newton polytopes). It introduces a generalized-diagonal machinery and uses Newton-polytope arguments to bound the Frobenius-orbit dimensions, avoiding inductive geometric arguments. The results yield effective, uniform bounds and have algorithmic consequences for computing diagonals modulo , with potential connections to geometric invariants like genus. Overall, the work sharpens the understanding of how diagonals of multivariate algebraic series behave modulo primes and provides practical, polynomial bounds on their algebraic degrees.

Abstract

In 1984, Deligne proved that for any prime number , the reduction modulo of the diagonal of a multivariate algebraic power series with integer coefficients is algebraic over the field of rational functions with coefficients in . Moreover, he conjectured that the algebraic degrees of these functions should grow at most polynomially in . In this article, we provide a new and elementary proof of Deligne's theorem, which yields the first general polynomial bound on with an explicit and reasonable degree.
Paper Structure (13 sections, 10 theorems, 57 equations)

This paper contains 13 sections, 10 theorems, 57 equations.

Table of Contents

  1. Introduction
  2. Comparison with previous works
  3. Organization of the article
  4. A sharper multivariate Christol theorem
  5. Preliminary results
  6. Section operators on $K(\!(T)\!)$
  7. A variant of Furstenberg's formula
  8. Section operators and residues
  9. Proof of \ref{['thm: main']}
  10. Diagonals
  11. An effective version of Deligne's theorem
  12. Generalized diagonals
  13. Proof of \ref{['thm:diag']}

Key Result

Theorem 1.1

Let $f \in \mathbb Z[\![\boldsymbol t]\!]$ be an algebraic power series with degree $d$, total height $h$, and partial height $(h_1,\ldots,h_n)$. Set Then, for every prime number $p$, $\Delta(f)_{|p}$ is annihilated by a linearized polynomial of $p$-degree at most $N$. In particular, $\Delta(f)_{|p}$ has degree at most $p^N-1$ over $\mathbb F_p(t)$.

Theorems & Definitions (23)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Theorem 1.4
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • Remark 2.4
  • Lemma 2.5
  • proof
  • ...and 13 more