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Solutions to the First Order Difference Equations in the Multivariate Difference Field

Lixin Du, Yarong Wei

Abstract

The bivariate difference field provides an algebraic framework for a sequence satisfying a recurrence of order two. Based on this, we focus on sequences satisfying a recurrence of higher order, and consider the multivariate difference field, in which the summation problem could be transformed into solving the first order difference equations. We then show a criterion for deciding whether the difference equation has a rational solution and present an algorithm for computing one rational solution of such a difference equation, if it exists. Moreover we get the rational solution set of such an equation.

Solutions to the First Order Difference Equations in the Multivariate Difference Field

Abstract

The bivariate difference field provides an algebraic framework for a sequence satisfying a recurrence of order two. Based on this, we focus on sequences satisfying a recurrence of higher order, and consider the multivariate difference field, in which the summation problem could be transformed into solving the first order difference equations. We then show a criterion for deciding whether the difference equation has a rational solution and present an algorithm for computing one rational solution of such a difference equation, if it exists. Moreover we get the rational solution set of such an equation.
Paper Structure (12 sections, 17 theorems, 98 equations)

This paper contains 12 sections, 17 theorems, 98 equations.

Table of Contents

  1. Introduction
  2. Multivariate Difference Field
  3. A difference isomorphism
  4. The constant field
  5. The spread computation
  6. The existence of rational solutions
  7. Decomposition of rational functions
  8. Orbital reduction for summability
  9. Reduction of the $c\sigma$-summability problem of rational functions in $V_{i}$
  10. Reduction of the $c\sigma$-summability problem of rational functions in $V_{[d]_{G,j}}$
  11. $c\sigma$-summability criterion for simple fractions
  12. The rational solution set

Key Result

Theorem 2.1

Let $( {\mathbb{F}}(\alpha_1,\alpha_2,\ldots,\alpha_n),\varphi)$ be a multivariate difference field with $\varphi|_{ {\mathbb{F}}}=\operatorname{id}$ and $(\varphi(\alpha_1,\alpha_2,\ldots,\alpha_n))=(\alpha_1,\alpha_2,\ldots,\alpha_n)A$, where $A$ is a diagonalizable matrix over ${\mathbb{F}}$ with

Theorems & Definitions (37)

  • Theorem 2.1
  • proof
  • Corollary 2.2
  • proof
  • Theorem 2.4
  • proof
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • proof
  • ...and 27 more