The authors discuss testing the relative primality of twopolynomials that are given with inexact coefficients or, in other words,testing whether two polynomials will always remain relatively prime (orcoprime) even if their coefficients are perturbed within a fixedperturbation bound. This is a more specialized problem than thecomputation of the greatest common divisor (gcd) of two polynomials withinexact coefficients. The latter problem has become quite popularlately, and the five references on it given by the authors should becomplemented by several others.

Though the known algorithms for the approximate gcd problems arequite effective (recall the algorithms based on the approximation ofpolynomial zeros and on the correlation of the gcd to Padéapproximation and to Toeplitz-Hankel computations in addition to the twoapproaches--based on the optimization and Euclideantechniques--cited by the authors), this paper argues that testingnumerical coprimality is even simpler than the computation of numericalgcd. The authors’ approach relies on the formulation of coprimality interms of the nonsingularity of the associated Sylvester matrixS. The neighborhood of the inputpolynomials where they remain coprime is defined via the conditionnumber of S, for whose computation the authors propose anextension of the Gohberg-Semencul formula to express the inverse ofS explicitly. (Alternatively, they could havespecialized the known formula for the inverse of more generalToeplitz-related matrices to their case of Sylvester matrices.) Thepaper is an interesting example of numerical/algebraic algorithms, whichare a major topic in computer algebra.