Approximate greatest common divisor of many polynomials and generalised resultants

Abstract

In this paper, a new characterisation of the approximate GCD of many polynomials is given that also allows the evaluation of accuracy of the corresponding ‘approximate GCD computation’. This new approach is based on some recent results on the factorisation of the generalised resultant of a set of polynomials into reduced resultants and appropriate Toeplitz matrices representing the exact GCD [1]. This allows the reduction of ‘approximate GCD’ computation to an equivalent ‘approximate factorisation’ of generalised resultants. This new approach may be formulated as a structured optimization problem (distance between structured matrices). We use this new framework to evaluate the ‘accuracy’ of the ‘approximate GCD’ of a certain degree. This evaluation is equivalent to finding the minimal perturbation on the original set of polynomials, which make the selected given degree ‘approximate GCD’ exact for the perturbed set. The later makes precise the meaning of approximate GCD, since it relates it to the exact notion on a perturbed set.