Probabilistic algorithms for computing resultants

Abstract

Let A and B be two polynomials in ℤ [x,y] and let R = resx(A,B) denote the resultant of A and B taken wrt x. In this paper we modify Collins' modular algorithm for computing R to make it output sensitive. The advantage of our algorithm is that it will be faster when the bounds needed by Collins' algorithm for the coefficients of R and for the degree of R are inaccurate. Our second contribution is an output sensitive modular algorithm for computing the monic resultant in ℚ[y]. The advantage of this algorithm is that it is faster still when the resultant has a large integer content. Both of our algorithms are necessarily probabilistic.The paper includes a number of resultant problems that motivate the need to consider such algorithms. We have implemented our algorithms in Maple. We have also implemented Collins' algorithm and the subresultant algorithm in Maple for comparison. The timings we obtain demonstrate that a good speedup is obtained.