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Solving Order 3 Difference Equations

Heba Bou KaedBey, Mark van Hoeij, Man Cheung Tsui

TL;DR

This paper resolves the problem of classifying order $3$ linear difference operators over $\, ext{and}$ the field $\,oldsymbol{C}(x)$ that are solvable in terms of lower-order operators. It develops a difference-module framework together with difference Galois theory, and introduces $2$-expressible sequences linked to Eulerian Galois groups to characterize solvability. The main result shows that a $2$-solvable order-$3$ operator must fall into one of three cases: reducible; gauge equivalent to $\,oldsymbol{\\Phi^{3}+f}$; or gauge equivalent to $L_{2}^{\circledS 2}\circ\circledS L_{1}$ with orders $1$ and $2$, with all cases being algorithmically solvable. The work generalizes known differential-case results to the difference setting, provides tools for higher-order analysis via restriction/induction and symmetric powers, and outlines a path toward a complete theory of $d$-solvability for difference modules.

Abstract

We classify order $3$ linear difference operators over $\mathbb{C}(x)$ that are solvable in terms of lower order difference operators. To prove this result, we introduce the notion of absolute irreducibility for difference modules, and classify (for arbitrary order) modules that are irreducible but not absolutely irreducible.

Solving Order 3 Difference Equations

TL;DR

This paper resolves the problem of classifying order linear difference operators over the field that are solvable in terms of lower-order operators. It develops a difference-module framework together with difference Galois theory, and introduces -expressible sequences linked to Eulerian Galois groups to characterize solvability. The main result shows that a -solvable order- operator must fall into one of three cases: reducible; gauge equivalent to ; or gauge equivalent to with orders and , with all cases being algorithmically solvable. The work generalizes known differential-case results to the difference setting, provides tools for higher-order analysis via restriction/induction and symmetric powers, and outlines a path toward a complete theory of -solvability for difference modules.

Abstract

We classify order linear difference operators over that are solvable in terms of lower order difference operators. To prove this result, we introduce the notion of absolute irreducibility for difference modules, and classify (for arbitrary order) modules that are irreducible but not absolutely irreducible.
Paper Structure (12 sections, 15 theorems, 37 equations, 1 table)

This paper contains 12 sections, 15 theorems, 37 equations, 1 table.

Table of Contents

  1. Introduction
  2. Background
  3. Difference modules
  4. Difference Galois theory
  5. 2-expressible sequences
  6. Symmetric powers
  7. Restriction and induction
  8. Examples of absolute (ir)reducibility
  9. Motivation of the Paper
  10. Semisimplicity and absolute irreducibility
  11. Main theorem
  12. A Gröbner basis computation

Key Result

Theorem 1

Let $L$ be an order $3$ difference operator over $\mathbb{C}(x)$. If $L$ is $2$-solvable (Definition solv2), then at least one of the following holds.

Theorems & Definitions (40)

  • Theorem : \ref{['thm:general-classification']} restated
  • Proposition 2.1: hendricks1999solving
  • Proposition 2.2
  • proof
  • Remark 3.1
  • Definition 3.2
  • Definition 3.3
  • Definition 3.4
  • Theorem 3.5
  • proof
  • ...and 30 more