Approximate greatest common divisor of polynomials and the structured singular value

In this note the following problem is considered: Given two monic coprime polynomials a(s) and b(s) with real coefficients, find the smallest (in magnitude) perturbation in their coefficients so 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, which can be performed using efficient existing techniques of robust control. A simple numerical example illustrates the effectiveness of the method. The generalisation of the method to calculate the approximate greatest common divisor (GCD) of polynomials is finally discussed.