Superfast solution of Toeplitz systems based on syzygy reduction

We present a new superfast algorithm for solving Toeplitz systems. This algorithm is based on a relation between the solution of such problems and 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 and the solution of a Toeplitz system T u = g can be reinterpreted as the remainder of a vector depending on g, by these two generators. We obtain these generators and this remainder with computational complexity O(n log n) for a Toeplitz matrix of size n× n.