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Parameterized D-torsors in differential Galois theory

Omar León Sánchez, David Meretzky

Abstract

In the context of differential fields of characteristic zero with several commuting derivations, we discuss the notion of $\#$-differential equations on parameterized D-torsors and their associated Galois extensions. Using model-theoretic methods, we observe that any generalized strongly normal extension (in the sense of Pillay [14] and, more generally, León Sánchez [9]) is the Galois extension of a parameterized D-torsor. Furthermore, we prove a parameterized version of a theorem of Kolchin on differential cohomology, itself of independent interest, and use it to provide a necessary and sufficient cohomological condition for when a generalized strongly normal extension is the Galois extension for a log-differential equation on its Galois group (as a parameterized D-group). We also present general model-theoretic versions of some of the main results.

Parameterized D-torsors in differential Galois theory

Abstract

In the context of differential fields of characteristic zero with several commuting derivations, we discuss the notion of -differential equations on parameterized D-torsors and their associated Galois extensions. Using model-theoretic methods, we observe that any generalized strongly normal extension (in the sense of Pillay [14] and, more generally, León Sánchez [9]) is the Galois extension of a parameterized D-torsor. Furthermore, we prove a parameterized version of a theorem of Kolchin on differential cohomology, itself of independent interest, and use it to provide a necessary and sufficient cohomological condition for when a generalized strongly normal extension is the Galois extension for a log-differential equation on its Galois group (as a parameterized D-group). We also present general model-theoretic versions of some of the main results.
Paper Structure (5 sections, 17 theorems, 35 equations)

This paper contains 5 sections, 17 theorems, 35 equations.

Table of Contents

  1. Introduction
  2. Some model-theoretic generalities
  3. Parameterized D-torsors and their differential Galois extensions
  4. A parameterized version of Kolchin's cohomology theorem
  5. The main result

Key Result

Theorem 1.2

Kolchin1 Let $\mathcal{U}^\Pi$ and $K^\Pi$ denote the $\Pi$-constants of $\mathcal{U}$ and $K$, respectively.

Theorems & Definitions (36)

  • Remark 1.1
  • Theorem 1.2
  • Proposition 1.3
  • Proposition 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Remark 1.7
  • Proposition 1.8
  • Theorem 1.9
  • Theorem 1.10: Main result
  • ...and 26 more