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Splitting differential equations using Galois theory

Christine Eagles, Léo Jimenez

TL;DR

The paper addresses whether pullbacks under the logarithmic differential preserve internality to the constants in differentially closed fields. It develops a general framework tying internality of pullbacks to the splitting of short exact sequences of binding groups via model-theoretic Galois theory and definable groupoids in $\omega$-stable theories, complemented by a definable Jordan decomposition for linear solvable groups. A key result shows that, when the base binding group is linear-nilpotent and the fibers have diagonalizable binding groups, almost internality of the total type is equivalent to almost splitting of the fibration; this leads to automatic splitting in nilpotent cases and provides an algebraic criterion for splitting. The authors also demonstrate non-splitting pullbacks in the elliptic-curve binding-group case, yielding explicit counterexamples to uniform internality and answering a question of Jin and Moosa. Overall, the work deepens the connection between model-theoretic Galois theory, groupoid retractability, and differential-algebraic internality, and provides effective criteria for the internality of logarithmic-differential pullbacks with significant implications for solving differential equations via finite data.

Abstract

This article is interested in pullbacks under the logarithmic derivative of algebraic ordinary differential equations. In particular, assuming the solution set of an equation is internal to the constants, we would like to determine when its pullback is itself internal to the constants. To do so, we develop, using model-theoretic Galois theory and differential algebra, a connection between internality of the pullback and the splitting of a short exact sequence of algebraic Galois groups. We then use algebraic group theory to obtain internality and non-internality results.

Splitting differential equations using Galois theory

TL;DR

The paper addresses whether pullbacks under the logarithmic differential preserve internality to the constants in differentially closed fields. It develops a general framework tying internality of pullbacks to the splitting of short exact sequences of binding groups via model-theoretic Galois theory and definable groupoids in -stable theories, complemented by a definable Jordan decomposition for linear solvable groups. A key result shows that, when the base binding group is linear-nilpotent and the fibers have diagonalizable binding groups, almost internality of the total type is equivalent to almost splitting of the fibration; this leads to automatic splitting in nilpotent cases and provides an algebraic criterion for splitting. The authors also demonstrate non-splitting pullbacks in the elliptic-curve binding-group case, yielding explicit counterexamples to uniform internality and answering a question of Jin and Moosa. Overall, the work deepens the connection between model-theoretic Galois theory, groupoid retractability, and differential-algebraic internality, and provides effective criteria for the internality of logarithmic-differential pullbacks with significant implications for solving differential equations via finite data.

Abstract

This article is interested in pullbacks under the logarithmic derivative of algebraic ordinary differential equations. In particular, assuming the solution set of an equation is internal to the constants, we would like to determine when its pullback is itself internal to the constants. To do so, we develop, using model-theoretic Galois theory and differential algebra, a connection between internality of the pullback and the splitting of a short exact sequence of algebraic Galois groups. We then use algebraic group theory to obtain internality and non-internality results.
Paper Structure (11 sections, 36 theorems, 42 equations)

This paper contains 11 sections, 36 theorems, 42 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Internality and binding groups
  4. Definable Jordan decomposition
  5. Splitting
  6. Definable fibrations and uniform internality
  7. Groupoids, splitting and short exact sequences
  8. Logarithmic-differential pullbacks in differentially closed fields
  9. Preliminaries on logarithmic-differential pullbacks
  10. Splitting when the binding group is nilpotent
  11. Non-splitting when the binding group is an elliptic curve

Key Result

Theorem A

Fix some $\omega$-stable theory $\mathrm{Th}$ eliminating imaginaries, some algebraically closed set of parameters $A$ and some $A$-definable pure algebraically closed field $\mathcal{C}$. Let $p \in S(A)$ be a weakly $\mathcal{C}$-orthogonal type and $f : p \rightarrow q$ be an $A$-definable map. A Then $p$ is almost $\mathcal{C}$-internal if and only if the map $f$ almost splits.

Theorems & Definitions (83)

  • Theorem A
  • Theorem B
  • Theorem C
  • Definition 2.1
  • Definition 2.3
  • proof
  • proof
  • Corollary 2.7
  • proof
  • Lemma 2.8
  • ...and 73 more