Certiied Approximate Univariate Gcds

We study the approximate GCD of two univariate polynomials given with limited accuracy or, equivalently, the exact GCD of the perturbed polynomials within some prescribed tolerance. A perturbed polynomial is regarded as a family of polynomials in a clas-siication space, which leads to an accurate analysis of the computation. Considering only the Sylvester matrix singular values, as is frequently suggested in the literature, does not suuce to solve the problem completely, even when the extended euclidean algorithm is also used. We provide a counterexample that illustrates this claim and indicates the problem's hardness. SVD computations on subre-sultant matrices lead to upper bounds on the degree of the approximate GCD. Further use of the subresultant matrices singular values yields an approximate syzygy of the given polynomials, which is used to establish a gap theorem on certain singular values that cer-tiies the maximum-degree approximate GCD. This approach leads directly to an algorithm for computing the approximate GCD polynomial. Lastly, we suggest the use of weighted norms in order to sharpen the theorem's conditions in a more intrinsic context.