When are Two Numerical Polynomials Relatively Prime?

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