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

2026-01-21

AI Summary

This study investigates the algebraicity of diagonals of multivariate algebraic power series modulo a prime $p$ and the growth behavior of their algebraic degrees $d_p$, aiming to verify Deligne’s conjecture that $d_p$ grows at most polynomially in $p$. By employing a constructive approach that integrates tools from algebraic power series, rational function fields over finite fields, reduction modulo $p$, and the theory of algebraic functions, the authors provide a new proof of Deligne’s theorem. Moreover, they establish—for the first time—an explicit polynomial upper bound on $d_p$ that holds in full generality. This result not only confirms a central aspect of Deligne’s conjecture but also offers a significantly more elementary and streamlined proof, thereby improving upon previous formulations.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Machine Learning: Deep Learning Theory
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

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.

Problem

• diagonals
• algebraicity modulo p
• degree bound
• Deligne's conjecture
• algebraic power series

Innovation

• diagonal of algebraic power series
• modular reduction
• algebraic degree bound
• Deligne's theorem
• polynomial growth