Nearest common root of polynomials, approximate greatest common divisor and the structured singular value

In this paper the following problem is considered: given two coprime polynomials, find the smallest perturbation in the magnitude of their coefficients such that the perturbed polynomials have a common root. It is shown that the problem is equivalent to the calculation of the structured singular value of a matrix arising in robust control and a numerical solution to the problem is developed. A simple numerical example illustrates the effectiveness of the method for two polynomials of low degree. Finally, problems involving the calculation of the approximate greatest common divisor of univariate polynomials are considered, by proposing a generalization of the definition of the structured singular value involving additional rank constraints.