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

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.