Numerical Univariate Polynomial GCD

We formalize the notion of approximate GCD for univariate poly-nomials given with limited accuracy and then address the problem of its computation. Algebraic concepts are applied in order to provide a solid foundation for a numerical approach. We exhibit the limitations of the euclidean algorithm through experiments, show that existing methods only solve part of the problem and assert its worst-case complexity. A rigorous geometrical point of view is given in the parameter space of all input polynomials and SVD computations on subresultants are applied in order to derive upper bounds on the degree of the approximate GCD. Then, we establish a certiication theorem and state the conditions under which it determines the precise GCD degree.