Sylvester's Forgotten Form of the Resultant

It is well known that Euclid's algorithm for computing the greatest common divisor (gcd) of two integer numbers is more than two thousand years old and, as it turns out, it is the oldest known algorithm. Interest in computing a gcd of two polynomials first appeared only in the sixteenth century and the problem was solved by Simon Stevin [13] simply by applying Euclid's algorithm (for integers) to polynomials with integer coefficients. However, from the computational point of view, Euclid's algorithm applied to polynomials with integer coefficients is very inefficient because of the growth of coefficients that takes place and the eventual slowdown of computations. This growth of coefficients is due to the fact that the ring Z[x] is not a Euclidean domain, and hence, divisions (as we know them) cannot always be performed. For example, take the two polynomials px(x) = x -7x4-7 andp2(x) = 3x 7 which have very small coefficients. Observe that, over the integers, we cannot divide px(x) hyp2(x) (since 3 does not divide 1) and, hence, we have to introduce the concept of pseudo-division, which always yields a pseudo-quotient and pseudo-remainder. According to this process, we have to premultiply Pi(x) by the leading coefficient of p2(x) raised to the power 2 [that is, we premultiply Pi() by 9 = 3] and then apply our usual polynomial division algorithm. [Below we denote the leading coefficient (1c) of a polynomial p(x) by lc(p(x)) and its degree by deg(j?(x)).] In the general case where deg(pl(x)) = n, and deg(p2(x)) ~m, we premultiply px{x) by lc(p2(x))~. In this was we know for sure that all the polynomial divisions involved in the process of computing a greatest common divisor of px{x) and p2(x) will be carried out in Z[x]. That is, in general, we start with