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

TL;DR

This paper explores the relationship between Toeplitz matrices and polynomial syzygies, providing a new perspective on solving Toeplitz systems by linking matrix generators to syzygy modules.

Contribution

It establishes an explicit connection between Toeplitz matrix generators and polynomial syzygies, offering a novel algebraic approach to solve Toeplitz systems.

Findings

Toeplitz matrices are linked to syzygy modules generated by two elements.

The solution to Toeplitz systems can be obtained as a remainder in a polynomial division.

The approach provides a new algebraic interpretation of Toeplitz system solutions.

Abstract

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.