Toeplitz and Toeplitz-block-Toeplitz matrices and their correlation with syzygies of polynomials

Abstract

In this paper, we re-investigate the resolution of Toeplitz systems $Tu=g$, from a new point of view, by correlating the solution of such problems with syzygies of polynomials or moving lines. We show an explicit connection between the generators of a Toeplitz matrix and the generators of the corresponding module of syzygies. We show that this module is generated by two elements of degree $n$ and the solution of $Tu=g$ can be reinterpreted as the remainder of an explicit vector depending on $g$, by these two generators.