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Positive characteristic Ax-Schanuel

Piotr Kowalski

TL;DR

This expository article surveys Ax-Schanuel type statements across characteristic zero and positive characteristic, framing the problem around the interplay of analytic, differential, and formal maps. It presents the classical characteristic-zero landscape (Schanuel’s conjectures, Ax’s differential theorem, and the dimension-intersection results) and then introduces the first positive-characteristic Ax-Schanuel result for additive power series, along with a broad conjectural framework using formal maps and good one-dimensional groups. The piece outlines two major directions for unifying Ax-Schanuel phenomena: a general connection-based formulation in characteristic zero (and its potential positive-characteristic analogues) and the development of Hasse-Schmidt differential Ax-Schanuel statements in positive characteristic. Collectively, the work highlights potential Diophantine and model-theoretic applications and charts pathways for extending Ax-Schanuel theory beyond characteristic zero through formal and HS-derivation methods. The article also situates these ideas within the broader context of Drinfeld modules, modular Ax-Schanuel, and ongoing research into general conjectures governing Ax-like transcendence phenomena.

Abstract

This expository paper is written in celebration of Boris Zilber's 75th birthday. We discuss Ax-Schanuel type statements focusing on the case of positive characteristic.

Positive characteristic Ax-Schanuel

TL;DR

This expository article surveys Ax-Schanuel type statements across characteristic zero and positive characteristic, framing the problem around the interplay of analytic, differential, and formal maps. It presents the classical characteristic-zero landscape (Schanuel’s conjectures, Ax’s differential theorem, and the dimension-intersection results) and then introduces the first positive-characteristic Ax-Schanuel result for additive power series, along with a broad conjectural framework using formal maps and good one-dimensional groups. The piece outlines two major directions for unifying Ax-Schanuel phenomena: a general connection-based formulation in characteristic zero (and its potential positive-characteristic analogues) and the development of Hasse-Schmidt differential Ax-Schanuel statements in positive characteristic. Collectively, the work highlights potential Diophantine and model-theoretic applications and charts pathways for extending Ax-Schanuel theory beyond characteristic zero through formal and HS-derivation methods. The article also situates these ideas within the broader context of Drinfeld modules, modular Ax-Schanuel, and ongoing research into general conjectures governing Ax-like transcendence phenomena.

Abstract

This expository paper is written in celebration of Boris Zilber's 75th birthday. We discuss Ax-Schanuel type statements focusing on the case of positive characteristic.
Paper Structure (8 sections, 5 theorems, 38 equations)

This paper contains 8 sections, 5 theorems, 38 equations.

Table of Contents

  1. Introduction
  2. Characteristic zero
  3. Results
  4. Motivations and applications
  5. Positive characteristic
  6. Recent ideas and speculations
  7. Towards a general statement of Ax-Schanuel
  8. Hasse-Schmidt differential Ax-Schanuel

Key Result

Theorem 2.2

Let be an analytic epimorphism where $E$ is an elliptic curve. Let $x_1,\dots ,x_n\in 1+X{\mathds{C}}\llbracket X\rrbracket$ be multiplicatively independent. Then

Theorems & Definitions (9)

  • Remark 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Remark 2.4
  • Definition 2.5: Bochner Boc
  • Theorem 3.1
  • Theorem 3.2
  • Example 3.3
  • Theorem 3.4