Parallel computation of bivariate polynomial resultants on graphics processing units

Abstract

Polynomial resultants are of fundamental importance in symbolic computations, especially in the field of quantifier elimination. In this paper we show how to compute the resultant res y (f, g) of two bivariate polynomials f,g ∈ ℤ [x, y] on a CUDA-capable graphics processing unit (GPU). We achieve parallelization by mapping the bivariate integer resultant onto a sufficiently large number of univariate resultants over finite fields, which are then lifted back to the original domain. We point out, that the commonly proposed special treatment for so called unlucky homomorphisms is unnecessary and how this simplifies the parallel resultant algorithm. All steps of the algorithm are executed entirely on the GPU. Data transfer is only used for the input polynomials and the resultant. Experimental results show the considerable speedup of our implementation compared to host-based algorithms. © 2012 Springer-Verlag.