Abstract
Subresultants are one of the most fundamental tools in computer algebra. They are at the core of numerous algorithms including, but not limited to, polynomial GCD computations, polynomial system solving, and symbolic integration. When the subresultant chain of two polynomials is involved in a client procedure, not all polynomials of the chain, or not all coefficients of a given subresultant, may be needed. Based on that observation, this paper discusses different practical schemes, and their implementation, for efficiently computing subresultants. Extensive experimentation supports our findings.
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Abdeljaoued, J., Diaz-Toca, G.M., González-Vega, L.: Bezout matrices, subresultant polynomials and parameters. Appl. Math. Comput. 214(2), 588–594 (2009)
Asadi, M., et al.: Basic Polynomial Algebra Subprograms (BPAS) (version 1.791) (2021). http://www.bpaslib.org
Asadi, M., Brandt, A., Moir, R.H.C., Moreno Maza, M., Xie, Y.: Parallelization of triangular decompositions: techniques and implementation. J. Symb. Comput. (2021, to appear)
Becker, E., Mora, T., Grazia Marinari, M., Traverso, C.: The shape of the shape lemma. In: Proceedings of ISSAC 1994, pp. 129–133. ACM (1994)
Chen, C., Moreno Maza, M.: Algorithms for computing triangular decomposition of polynomial systems. J. Symb. Comput. 47(6), 610–642 (2012)
Covanov, S., Mohajerani, D., Moreno Maza, M., Wang, L.: Big prime field FFT on multi-core processors. In: Proceedings of the 2019 International Symposium on Symbolic and Algebraic Computation (ISSAC), pp. 106–113. ACM (2019)
Covanov, S., Moreno Maza, M.: Putting Fürer algorithm into practice. Technical report (2014). http://www.csd.uwo.ca/~moreno//Publications/Svyatoslav-Covanov-Rapport-de-Stage-Recherche-2014.pdf
Della Dora, J., Dicrescenzo, C., Duval, D.: About a new method for computing in algebraic number fields. In: Caviness, B.F. (ed.) EUROCAL 1985. LNCS, vol. 204, pp. 289–290. Springer, Heidelberg (1985). https://doi.org/10.1007/3-540-15984-3_279
Ducos, L.: Algorithme de Bareiss, algorithme des sous-résultants. Informatique Théorique et Applications 30(4), 319–347 (1996)
Ducos, L.: Optimizations of the subresultant algorithm. J. Pure Appl. Algebra 145(2), 149–163 (2000)
Filatei, A., Li, X., Moreno Maza, M., Schost, E.: Implementation techniques for fast polynomial arithmetic in a high-level programming environment. In: Proceedings of ISSAC, pp. 93–100 (2006)
von zur Gathen, J., Gerhard, J.: Modern Computer Algebra, 3rd edn. Cambridge University Press, Cambridge (2013)
Granlund, T., The GMP Development Team: GNU MP: the GNU multiple precision arithmetic library (version 6.1.2) (2020). http://gmplib.org
van der Hoeven, J., Lecerf, G., Mourrain, B.: Mathemagix (from 2002). http://www.mathemagix.org
Kahoui, M.E.: An elementary approach to subresultants theory. J. Symb. Comput. 35(3), 281–292 (2003)
Knuth, D.E.: The analysis of algorithms. Actes du congres international des Mathématiciens 3, 269–274 (1970)
Lecerf, G.: On the complexity of the Lickteig-Roy subresultant algorithm. J. Symb. Comput. 92, 243–268 (2019)
Lehmer, D.H.: Euclid’s algorithm for large numbers. Am. Math. Mon. 45(4), 227–233 (1938)
Lemaire, F., Moreno Maza, M., Xie, Y.: The RegularChains library in MAPLE. In: Maple Conference, vol. 5, pp. 355–368 (2005)
Lickteig, T., Roy, M.F.: Semi-algebraic complexity of quotients and sign determination of remainders. J. Complex. 12(4), 545–571 (1996)
Maplesoft, a division of Waterloo Maple Inc.: Maple (2020). www.maplesoft.com
Monagan, M.: Probabilistic algorithms for computing resultants. In: Proceedings of ISSAC 2005, pp. 245–252. ACM (2005)
Monagan, M., Tuncer, B.: Factoring multivariate polynomials with many factors and huge coefficients. In: Gerdt, V.P., Koepf, W., Seiler, W.M., Vorozhtsov, E.V. (eds.) CASC 2018. LNCS, vol. 11077, pp. 319–334. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-99639-4_22
Montgomery, P.L.: Modular multiplication without trial division. Math. Comput. 44(170), 519–521 (1985)
Reischert, D.: Asymptotically fast computation of subresultants. In: Proceedings of ISSAC 1997, pp. 233–240. ACM (1997)
Schönhage, A.: Schnelle Berechnung von Kettenbruchentwicklungen. Acta Informatica 1, 139–144 (1971)
Shoup, V., et al.: NTL: a library for doing number theory (version 11.4.3) (2021). www.shoup.net/ntl
Thull, K., Yap, C.: A unified approach to HGCD algorithms for polynomials and integers. Manuscript (1990)
Zippel, R.: Probabilistic algorithms for sparse polynomials. In: Ng, E.W. (ed.) Symbolic and Algebraic Computation. LNCS, vol. 72, pp. 216–226. Springer, Heidelberg (1979). https://doi.org/10.1007/3-540-09519-5_73
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The authors would like to thank Robert H. C. Moir and NSERC of Canada (award CGSD3-535362-2019).
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A Maple code for Polynomial Systems
A Maple code for Polynomial Systems
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Asadi, M., Brandt, A., Moreno Maza, M. (2021). Computational Schemes for Subresultant Chains. In: Boulier, F., England, M., Sadykov, T.M., Vorozhtsov, E.V. (eds) Computer Algebra in Scientific Computing. CASC 2021. Lecture Notes in Computer Science(), vol 12865. Springer, Cham. https://doi.org/10.1007/978-3-030-85165-1_3
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