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
This is a preview of subscription content, access via your institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Bubeck, T., Hiller, M., Küchlin, W., Rosenstiel, W.: Distributed Symbolic Computation with DTS. In: Ferreira, A., Rolim, J.D.P. (eds.) IRREGULAR 1995. LNCS, vol. 980, pp. 231–248. Springer, Heidelberg (1995)
Collins, G.E.: The calculation of multivariate polynomial resultants. In: SYMSAC 1971: Proceedings of the Second ACM Symposium on Symbolic and Algebraic Manipulation, pp. 212–222. ACM, New York (1971)
Czapor, S., Geddes, K., Labahn, G.: Algorithms for Computer Algebra. Kluwer Academic Publishers (1992)
Graham, R.L., Knuth, D.E., Patashnik, O.: Concrete Mathematics: A Foundation for Computer Science. Addison-Wesley Longman Publishing Co., Inc., Boston (1990)
Hong, H., Loidl, H.W.: Parallel Computation of Modular Multivariate Polynomial Resultants on a Shared Memory Machine. In: Buchberger, B., Volkert, J. (eds.) CONPAR 1994 and VAPP 1994. LNCS, vol. 854, pp. 325–336. Springer, Heidelberg (1994), http://www.springerlink.com/content/fp43323vv51104q6/
Knuth, D.E.: The Art of Computer Programming: Seminumerical Algorithms, 3rd edn., vol. 2. Addison-Wesley Professional (1997)
Winkler, F.: Polynomial Algorithms in Computer Algebra. Springer-Verlag New York, Inc., Secaucus (1996)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
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
Download citation
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)
