Bézout Identities Associated to a Finite Sequence

We consider finite sequences s ∈ 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 Bézout identities. We give a number of applications. Firstly, a new (quadratic) algorithm to compute Bézout coefficients over a field, which compares favourably with the extended Euclidean algorithm. We show that the successive output polynomials of the Berlekamp-Massey algorithm either coincide or are relatively prime. A second application concerns sequences with perfect linear complexity profile. We give new characterisations of them in terms of minimal polynomials and a simpler proof of a theorem of the characterisation of binary sequences with perfect linear complexity due to Wang and Massey using the third set of identities. Another application is to annihilating polynomials which do not vanish at zero and have minimal degree. We simplify and extend an algorithm of Salagean to sequences over D. First we prove a lower bound lemma which was stated without proof for sequences over a field. This lemma and an easy extension of the author's minimal polynomial algorithm yields such annihilators. The first set of identities allows us to remove a test. In fact, we compute minimal realisations and give corresponding identities. We also construct these annihilators by extending the sequence by one term, give corresponding minimal polynomial identities and prove a characterisation (stated without proof by Salagean over a field). In the Appendix, we give an alternative proof of the lower bound lemma using reciprocal annihilators and apply it to the complexity of reverse sequences. This gives 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.