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The theory DCF$_p$A exists for $p>0$

Kai Ino, Omar Leon Sanchez

TL;DR

This paper proves the existence of a model companion for differential-difference fields in characteristic $p>0$, denoted $DCF_pA$. By developing the dd-kernel and dd-realisation machinery and leveraging a result of Chatidakis-Chatzidakis on generic automorphisms, it provides a complete, model-complete, and stable theory extending $DCF_p$ with a generic automorphism. The work also offers alternative first-order axiomatisations for $DCF$ and $DCF_0A$, clarifying the landscape of companions in positive characteristic and highlighting the role of separable vs. inseparable phenomena. The results connect to Chatzidakis-Pillay’s framework and raise directions for extensions to multiple derivations and separably closed settings.

Abstract

We prove that the (elementary) class of differential-difference fields in characteristic $p>0$ admits a model-companion. In the terminology of Chatzidakis-Pillay, this says that the class of differentially closed fields of characteristic $p$ equipped with a generic differential-automorphism is elementary; i.e., DCF$_p$A exists. Along the way, we provide alternative first-order axiomatisations for DCF (differentially closed fields) and also for DCF$_0$A.

The theory DCF$_p$A exists for $p>0$

TL;DR

This paper proves the existence of a model companion for differential-difference fields in characteristic , denoted . By developing the dd-kernel and dd-realisation machinery and leveraging a result of Chatidakis-Chatzidakis on generic automorphisms, it provides a complete, model-complete, and stable theory extending with a generic automorphism. The work also offers alternative first-order axiomatisations for and , clarifying the landscape of companions in positive characteristic and highlighting the role of separable vs. inseparable phenomena. The results connect to Chatzidakis-Pillay’s framework and raise directions for extensions to multiple derivations and separably closed settings.

Abstract

We prove that the (elementary) class of differential-difference fields in characteristic admits a model-companion. In the terminology of Chatzidakis-Pillay, this says that the class of differentially closed fields of characteristic equipped with a generic differential-automorphism is elementary; i.e., DCFA exists. Along the way, we provide alternative first-order axiomatisations for DCF (differentially closed fields) and also for DCFA.

Paper Structure

This paper contains 9 sections, 18 theorems, 86 equations.

Table of Contents

  1. Introduction
  2. Some preliminary results
  3. Two remarks on separable extensions
  4. On separable differential extensions
  5. On separable difference extensions
  6. Differential kernels
  7. $dd$-kernels and $dd$-realisations
  8. On $dd$-realisations in characteristic $p>0$
  9. The model-companion DCF$_p$A

Key Result

Lemma 2.1

Jacobson64 Let $L/K$ be a field extension and $\delta:K\to L$ a derivation.

Theorems & Definitions (49)

  • Lemma 2.1
  • Remark 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Example 2.6
  • Proposition 2.7
  • proof
  • Remark 2.8
  • ...and 39 more