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On Recurrence Relations of Multi-dimensional Sequences

Hamid Rahkooy

Abstract

In this paper, we present a new algorithm for computing the linear recurrence relations of multi-dimensional sequences. Existing algorithms for computing these relations arise in computational algebra and include constructing structured matrices and computing their kernels. The challenging problem is to reduce the size of the corresponding matrices. In this paper, we show how to convert the problem of computing recurrence relations of multi-dimensional sequences into computing the orthogonal of certain ideals as subvector spaces of the dual module of polynomials. We propose an algorithm using efficient dual module computation algorithms. We present a complexity bound for this algorithm, carry on experiments using Maple implementation, and discuss the cases when using this algorithm is much faster than the existing approaches.

On Recurrence Relations of Multi-dimensional Sequences

Abstract

In this paper, we present a new algorithm for computing the linear recurrence relations of multi-dimensional sequences. Existing algorithms for computing these relations arise in computational algebra and include constructing structured matrices and computing their kernels. The challenging problem is to reduce the size of the corresponding matrices. In this paper, we show how to convert the problem of computing recurrence relations of multi-dimensional sequences into computing the orthogonal of certain ideals as subvector spaces of the dual module of polynomials. We propose an algorithm using efficient dual module computation algorithms. We present a complexity bound for this algorithm, carry on experiments using Maple implementation, and discuss the cases when using this algorithm is much faster than the existing approaches.

Paper Structure

This paper contains 11 sections, 9 theorems, 15 equations, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Annihilator of $n-$dimensional Sequences
  4. Dual Module of Polynomials
  5. Dual Module vs Annihilator
  6. The Algorithm
  7. Reciprocal of a Sequence
  8. The Algorithm
  9. Complexity and Experiments
  10. Complexity of the Algorithm
  11. Experiments

Key Result

Theorem 3

There is a one to one correspondence between finite dimensional subspaces of $\mathbb{K} [\partial]$ that are closed under derivation and $\mathfrak{m}\text{-primary}$ ideals in $\mathbb{K} [x_1,\dots,x_n]$, where $\mathfrak{m} = \left\langle x_1,\ldots,x_n\right\rangle$. Moreover, If $I$ is an $\ma

Theorems & Definitions (34)

  • Definition 1: $n-$dimensional Sequence bms-neiger-rahkooy-schost-casc
  • Definition 2: Annihilator of a Sequence bms-neiger-rahkooy-schost-casc, Section 2.1
  • Theorem 3: Mora, et al. 1993, mari-mora-moel-mult, Theorem 3.1
  • Theorem 4: Mourrain, 1997, mourrain1997
  • Proposition 5: Proposition 4.1, mourrain1997
  • Definition 6
  • Remark 7
  • Remark 8
  • Remark 9
  • Remark 10
  • ...and 24 more