Computing approximate GCD of univariate polynomials

In this paper we discuss the problem of computing the approximate GCD of two univariate polynomials. We construct a linearly structured resultant matrix from given polynomials. We show the equivalence of the full rank property of this resultant matrix and the coprimeness of the polynomials. Further we show that the nearest structured low rank approximation (SLRA) of the resultant matrix gives the approximate GCD of the polynomials. We formulate the problem of computing the nearest SLRA as an optimization problem on a smooth manifold, namely the unit sphere S^N-1 in ℝ^N.