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Parallel Computation of Bivariate Polynomial Resultants on Graphics Processing Units

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Part of the Lecture Notes in Computer Science book series (LNTCS,volume 7134)

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 \(\ensuremath\operatorname{res}_y(f,g)\) of two bivariate polynomials \(f,g\in\ensuremath\mathbb{Z}[x,y]\) 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.

Keywords

  • polynomial resultants
  • modular algorithm
  • parallelization
  • GPU
  • CUDA
  • graphics hardware
  • symbolic computation

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Stussak, C., Schenzel, P. (2012). Parallel Computation of Bivariate Polynomial Resultants on Graphics Processing Units. In: Jónasson, K. (eds) Applied Parallel and Scientific Computing. PARA 2010. Lecture Notes in Computer Science, vol 7134. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-28145-7_8

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  • DOI: https://doi.org/10.1007/978-3-642-28145-7_8

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-28144-0

  • Online ISBN: 978-3-642-28145-7

  • eBook Packages: Computer ScienceComputer Science (R0)