Approximate Greatest Common Divisors of polynomials and the optimal solution

The Greatest Common Divisor (GCD) of many polynomials is central to linear systems problems and its computation is a nongeneric problem. Defining the notion of “approximate” GCD, measuring and computing the strength of the approximation and determining the “best approximation” are challenging problems. This paper uses the Sylvester Resultant representation of the GCD of many polynomials, and the corresponding factorisation of generalised resultants. We define the notion of “approximate GCD” and then indicate how to compute the “optimal approximate GCD” of a given order, or degree and the corresponding order of the approximation. This optimisation problem is defined as a distance problem in a projective space and it is shown to have an analytic solution.