Toeplitz and Toeplitz-block-Toeplitz matrices and their correlation with syzygies of polynomials
For researchers in numerical linear algebra and symbolic computation, this provides a new theoretical perspective on Toeplitz systems, but the results are theoretical and lack concrete performance numbers.
This paper establishes an explicit connection between Toeplitz matrix generators and syzygy modules, showing that the solution of a Toeplitz system can be reinterpreted as a remainder of an explicit vector by two generators of degree n.
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.