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Computing differential Galois groups of second-order linear $q$-difference equations

Carlos E. Arreche, Yi Zhang

TL;DR

The paper develops an algorithmic framework to compute the differential Galois group $G$ for second-order linear $q$-difference equations with rational coefficients by integrating Hardouin–Singer differential Galois theory with Hendriks’ difference Galois approach. The strategy is to first determine the difference Galois group $H$ over an extended base field via Hendriks’ algorithm, then extract the differential-algebraic constraints defining $G$ as a subgroup of $H$, accounting for unipotent radicals and various primitive/imprimitive, reducible, or irreducible cases. It provides explicit criteria (via Riccati equations, $q$-discrete residues, and gauge transformations) to classify $G$ into diagonalizable, reducible non-diagonalizable, irreducible imprimitive (three subtypes), or irreducible primitive, including how to compute the defining equations in each case. The methodology is illustrated on concrete examples, including equations satisfied by knot-related colored Jones polynomials, and yields explicit descriptions of $G$ (and in particular its possible $G_a$ or $G_m$-type components and determinants). This work extends the differential Galois toolkit for difference equations and sets a blueprint for higher-order and elliptic-curve settings.

Abstract

We apply the differential Galois theory for difference equations developed by Hardouin and Singer to compute the differential Galois group for a second-order linear $q$-difference equation with rational function coefficients. This Galois group encodes the possible polynomial differential relations among the solutions of the equation. We apply our results to compute the differential Galois groups of several concrete $q$-difference equations, including for the colored Jones polynomial of a certain knot.

Computing differential Galois groups of second-order linear $q$-difference equations

TL;DR

The paper develops an algorithmic framework to compute the differential Galois group for second-order linear -difference equations with rational coefficients by integrating Hardouin–Singer differential Galois theory with Hendriks’ difference Galois approach. The strategy is to first determine the difference Galois group over an extended base field via Hendriks’ algorithm, then extract the differential-algebraic constraints defining as a subgroup of , accounting for unipotent radicals and various primitive/imprimitive, reducible, or irreducible cases. It provides explicit criteria (via Riccati equations, -discrete residues, and gauge transformations) to classify into diagonalizable, reducible non-diagonalizable, irreducible imprimitive (three subtypes), or irreducible primitive, including how to compute the defining equations in each case. The methodology is illustrated on concrete examples, including equations satisfied by knot-related colored Jones polynomials, and yields explicit descriptions of (and in particular its possible or -type components and determinants). This work extends the differential Galois toolkit for difference equations and sets a blueprint for higher-order and elliptic-curve settings.

Abstract

We apply the differential Galois theory for difference equations developed by Hardouin and Singer to compute the differential Galois group for a second-order linear -difference equation with rational function coefficients. This Galois group encodes the possible polynomial differential relations among the solutions of the equation. We apply our results to compute the differential Galois groups of several concrete -difference equations, including for the colored Jones polynomial of a certain knot.

Paper Structure

This paper contains 16 sections, 24 theorems, 173 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on differential Galois theory for difference equations
  3. Differential relations among solutions of first-order $q$-difference equations
  4. Hendriks' algorithm
  5. Diagonalizable groups
  6. Reducible non-diagonalizable groups
  7. Irreducible and imprimitive groups
  8. Irreducible and imprimitive (1): diagonalizable over the quadratic extension
  9. Irreducible and imprimitive (2): rational system of imprimitivity
  10. Irreducible and imprimitive (3): quadratic system of imprimitivity
  11. Irreducible and primitive groups
  12. Examples
  13. Example
  14. Example
  15. Example
  16. ...and 1 more sections

Key Result

Theorem 2.4

(Cf. hardouin-singer:2008) If $k^\sigma=C$ is $\delta$-closed, there exists a $\sigma\delta$-PV ring for difeq1, and it is unique up to $\sigma\delta$-$k$-isomorphism. Moreover, $R^\sigma=k^\sigma$.

Theorems & Definitions (49)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.4
  • Proposition 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Proposition 2.9
  • Theorem 2.10
  • ...and 39 more