This paper examines the computation of polynomial greatest common divisors by various generalizations of Euclid's algorithm. The phenomenon of coefficient growth is described, and the history of successful efforts first to control it and then to eliminate it is related.
The recently developed modular algorithm is presented in careful detail, with special attention to the case of multivariate polynomials.
The computing times for the subresultant PRS algorithm, which is essentially the best of its kind, and for the modular algorithm are analyzed, and it is shown that the modular algorithm is markedly superior. In fact, the modular algorithm can obtain a GCD in less time than is required to verify it by classical division.