The null space of the Bezout matrix in any basis and gcd's

TL;DR

This paper generalizes the understanding of the null space of the Bezout matrix to any polynomial basis and introduces two basis-independent methods for computing polynomial GCDs using Bezout matrices.

Contribution

It extends the null space structure of the Bezout matrix from the monomial basis to arbitrary bases and proposes basis-independent GCD computation methods.

Findings

Null space structure is valid in any basis.

Two new GCD algorithms using Bezout matrices are introduced.

Results applicable to various polynomial bases in practice.

Abstract

This manuscript presents a generalization of the structure of the null space of the Bezout matrix in the monomial basis, see [G. Heinig and K. Rost, Algebraic methods for toeplitz-like matrices and operators, 1984], to an arbitrary basis. In addition, two methods for computing the gcd of several polynomials, using also Bezout matrices, without having to convert them to the monomial basis. The main point is that the presented results are expressed with respect to an arbitrary polynomial basis. In recent years, many problems in polynomial systems, stability theory, CAGD, etc., are solved using Bezout matrices in distinct specific bases. Therefore, it is very useful to have results and tools that can be applied to any basis.