B\'{e}zout Identities Associated to a Finite Sequence

TL;DR

This paper introduces new Bézout identities linked to finite sequences over integral domains, providing algorithms and applications such as computing Bézout coefficients, analyzing sequence complexity, and extending existing algorithms.

Contribution

It presents three sets of Bézout identities for finite sequences over integral domains, along with quadratic algorithms and multiple applications including sequence analysis and polynomial computation.

Findings

Identifies three sets of Bézout identities involving sequence-associated polynomials.

Provides quadratic complexity algorithms for computing sequence coefficients.

Extends and simplifies algorithms for sequence analysis over factorial domains.

Abstract

We consider finite sequences $s\in D^n$ where $D$ is a commutative, unital, integral domain. We prove three sets of identities (possibly with repetitions), each involving $2n$ polynomials associated to $s$. The right-hand side of these identities is a recursively-defined (non-zero) 'product-of-discrepancies'. There are implied iterative algorithms (of quadratic complexity) for the left-hand side coefficients; when the ground domain is factorial, the identities are in effect B\'ezout identities. We give a number of applications: an algorithm to compute B\'ezout coefficients over a field; the outputs of the Berlekamp-Massey algorithm; sequences with perfect linear complexity profile; annihilating polynomials which do not vanish at zero and have minimal degree: we simplify and extend an algorithm of Salagean to sequences over $D$. In the Appendix, we give a new proof of a theorem of Imamura and Yoshida on the linear complexity of reverse sequences, initially proved using Hankel matrices over a field and now valid for sequences over a factorial domain.