Shortest Two-way Linear Recurrences

TL;DR

This paper introduces a simpler algorithm for computing shortest two-way linear recurrences in finite sequences, improving on previous methods by leveraging polynomial properties and degree bounds.

Contribution

The authors present a new, simplified algorithm for finding shortest two-way linear recurrences, demonstrating polynomial coprimality and combining degree bounds with existing algorithms.

Findings

The pairs of polynomials used are always relatively prime.

The new algorithm is simpler and more direct than previous methods.

The increase in shortest length is bounded by max{n+1-2*LC_n, 0}.

Abstract

Let $s$ be a finite sequence over a field of length $n$. It is well-known that if $s$ satisfies a linear recurrence of order $d$ with non-zero constant term, then the reverse of $s$ also satisfies a recurrence of order $d$ (with coefficients in reverse order). A recent article of A. Salagean proposed an algorithm to find such a shortest 'two-way' recurrence -- which may be longer than a linear recurrence for $s$ of shortest length $\LC_n$. We give a new and simpler algorithm to compute a shortest two-way linear recurrence. First we show that the pairs of polynomials we use to construct a minimal polynomial iteratively are always relatively prime; we also give the extended multipliers. Then we combine degree lower bounds with a straightforward rewrite of a published algorithm due to the author to obtain our simpler algorithm. The increase in shortest length is $\max\{n+1-2\LC_n,0\}$.