When are Two Numerical Polynomials Relatively Prime?

Let a andb be two polynomials having numerical coefficients. We consider the question: When are a andb relatively prime? Since the coefficients of a and bare approximant, the question is the same as: When are two polynomials relatively prime, even after small perturbations of the coefficients? In this paper we provide a numeric parameter for determining whether two polynomials are prime, even under small perturbations of the coefficients. Our methods rely on an inversion formula for Sylvester matrices to establish an effective criterion for relative primeness. The inversion formula can also be used to approximate the condition number of a Sylvester matrix.