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On the computation of the difference Galois groups of order three equations

Thomas Dreyfus, Marina Poulet

TL;DR

The paper addresses the computation of difference Galois groups for order-3 equations across the shift, q-difference, Mahler, and elliptic cases, framing the problem within difference Galois theory and its connection to differential transcendence.It develops a unified strategy based on gauge transformations and reduced forms to distinguish reducible, imprimitive, and primitive Galois groups, and to reduce higher-order problems to order-1 and order-2 subproblems, with explicit criteria (notably Riccati-type conditions) and structural results for prime-degree systems.A key contribution is the detailed classification and computation scheme for order-3 equations, including an explicit running example from Wadim Zudilin, and an application that yields differential-transcendence conclusions under suitable hypotheses.The results provide both theoretical insights and practical reduction techniques, enabling determination of Galois groups (GL_3, SL_3, SO_3, or torus-based groups) and supporting differential-transcendence proofs in a broad class of difference equations.

Abstract

In this paper we consider the problem of computing the difference Galois groups of order three equations for a large class of difference operators including the shift operator (Case S), the $q$-difference operator (Case Q), the Mahler operator (Case M) and the elliptic case (Case E). We show that the general problem can be reduced to several ancillary problems. We prove criteria to detect the irreducible and imprimitive Galois groups. Finally, we give a sufficient condition of differential transcendence of solutions of order three difference equations. We also compute the difference Galois group of an equation suggested by Wadim Zudilin.

On the computation of the difference Galois groups of order three equations

TL;DR

The paper addresses the computation of difference Galois groups for order-3 equations across the shift, q-difference, Mahler, and elliptic cases, framing the problem within difference Galois theory and its connection to differential transcendence.It develops a unified strategy based on gauge transformations and reduced forms to distinguish reducible, imprimitive, and primitive Galois groups, and to reduce higher-order problems to order-1 and order-2 subproblems, with explicit criteria (notably Riccati-type conditions) and structural results for prime-degree systems.A key contribution is the detailed classification and computation scheme for order-3 equations, including an explicit running example from Wadim Zudilin, and an application that yields differential-transcendence conclusions under suitable hypotheses.The results provide both theoretical insights and practical reduction techniques, enabling determination of Galois groups (GL_3, SL_3, SO_3, or torus-based groups) and supporting differential-transcendence proofs in a broad class of difference equations.

Abstract

In this paper we consider the problem of computing the difference Galois groups of order three equations for a large class of difference operators including the shift operator (Case S), the -difference operator (Case Q), the Mahler operator (Case M) and the elliptic case (Case E). We show that the general problem can be reduced to several ancillary problems. We prove criteria to detect the irreducible and imprimitive Galois groups. Finally, we give a sufficient condition of differential transcendence of solutions of order three difference equations. We also compute the difference Galois group of an equation suggested by Wadim Zudilin.
Paper Structure (21 sections, 32 theorems, 80 equations, 1 figure)

This paper contains 21 sections, 32 theorems, 80 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Difference algebra
  3. Galois groups
  4. Factorisations of difference operators
  5. From theory to practice
  6. Galois groups of difference systems
  7. The reducible case
  8. The imprimitive case
  9. Imprimitive groups that may occur
  10. How to compute a reduced form?
  11. The primitive case
  12. Galois groups of equations of order one and diagonal systems
  13. Galois groups of equations of order one
  14. Galois groups of diagonal systems
  15. Galois groups of equations of order two
  16. ...and 6 more sections

Key Result

Theorem 1

For all $t\in \mathbb{C}^*$, the difference Galois group of eq0 is $\mathrm{GL}_3 (\mathbb{C})$.

Figures (1)

  • Figure 1: The Newton polygon of the operator associated with $U(z,t)$.

Theorems & Definitions (63)

  • Theorem : Theorem \ref{['thm2']}
  • Theorem 2.1
  • Definition 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5: Lemma 2 in abramov2021rational
  • Remark 2.6
  • Example 2.7
  • Lemma 3.1
  • proof
  • ...and 53 more