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

In this paper, we re-investigate the resolution of Toeplitz systems T u = 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 T u = g can be reinterpreted as the remainder of an explicit vector depending on g, by these two generators. This approach extends naturally to multivariate problems and we describe for Toeplitz-block-Toeplitz matrices, the structure of the corresponding generators.