Transcendence properties of the Artin–Hasse exponential modulo p

Kramer-Miller, Joe

doi:10.2140/ant.2026.20.1597

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Transcendence properties of the Artin–Hasse exponential modulo $p$

Joe Kramer-Miller

Vol. 20 (2026), No. 8, 1597–1614

DOI: 10.2140/ant.2026.20.1597

Abstract

Let $E_{p} (x)$ denote the Artin–Hasse exponential and ${\bar{E}}_{p} (x)$ denote its reduction modulo $p$ in $𝔽_{p} [[x]]$ . We study transcendence properties of ${\bar{E}}_{p} (x)$ over $𝔽_{p} [x]$ . We give two proofs that ${\bar{E}}_{p} (x)$ is transcendental, affirmatively answering a question of Thakur. We also prove algebraic independence results: (i) for $f_{1}, \dots, f_{r} \in x {\bar{𝔽}}_{p} [x]$ satisfying certain linear independence properties, we show that the ${\bar{E}}_{p} (f_{1}), \dots, {\bar{E}}_{p} (f_{r})$ are algebraically independent over $𝔽_{p} [x]$ ; and (ii) we determine the algebraic relations between the ${\bar{E}}_{p} (ξ x)$ for $ξ \in 𝔽_{p}^{\times}$ . Our proof studies the higher derivatives of ${\bar{E}}_{p} (x)$ and makes use of iterative differential Galois theory.

Keywords

transcendence in positive characteristic, differential Galois theory

Mathematical Subject Classification

Primary: 11B85, 11J91

Milestones

Received: 25 May 2024

Revised: 5 May 2025

Accepted: 27 June 2025

Published: 25 September 2026

Authors

Joe Kramer-Miller
Department of Mathematics Lehigh University Bethlehem, PA United States

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Vol. 20, No. 8, 2026