  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^2 n) for a Toeplitz
matrix of size nxn.
