Approximate greatest common divisor of many polynomials, generalised resultants, and strength of approximation

The computation of the greatest common divisor (GCD) of many polynomials is a nongeneric problem. Techniques defining “approximate GCD” solutions have been defined, but the proper definition of the “approximate” GCD, and the way we can measure the strength of the approximation has remained open. This paper uses recent results on the representation of the GCD of many polynomials, in terms of factorisation of generalised resultants, to define the notion of “approximate GCD” and define the strength of any given approximation by solving an optimisation problem. The newly established framework is used to evaluate the performance of alternative procedures which have been used for defining approximate GCDs.