Algorithms for Linearly Recurrent Sequences of Truncated Polynomials

TL;DR

This paper introduces three algorithms for finding recurrence relations in sequences of truncated polynomials, improving efficiency and demonstrating practical applications in algebraic computations.

Contribution

The paper presents novel complexity improvements for existing algorithms and applies them to compute recurrence relations in sequences over truncated polynomial rings.

Findings

Complexity improvements for Berlekamp-Massey-like and Hankel matrix kernel methods.

Empirical validation through C++ implementation.

Applications to minimal polynomials and determinants over truncated polynomial rings.

Abstract

Linear recurrent sequences are those whose elements are defined as linear combinations of preceding elements, and finding recurrence relations is a fundamental problem in computer algebra. In this paper, we focus on sequences whose elements are vectors over the ring $\mathbb{A} = \mathbb{K}[x]/(x^d)$ of truncated polynomials. Finding the ideal of their recurrence relations has applications such as the computation of minimal polynomials and determinants of sparse matrices over $\mathbb{A}$. We present three methods for finding this ideal: a Berlekamp-Massey-like approach due to Kurakin, one which computes the kernel of some block-Hankel matrix over $\mathbb{A}$ via a minimal approximant basis, and one based on bivariate Pad\'e approximation. We propose complexity improvements for the first two methods, respectively by avoiding the computation of redundant relations and by exploiting the Hankel structure to compress the approximation problem. Then we confirm these improvements empirically through a C++ implementation, and we discuss the above-mentioned applications.