Abstract
We discuss the design and implementation of multivariate power series, univariate polynomials over power series, and their associated arithmetic operations within the Basic Polynomial Algebra Subprograms (BPAS) Library. This implementation employs lazy variations of Weierstrass preparation and the factorization of univariate polynomials over power series following Hensel’s lemma. Our implementation is lazy in that power series terms are only computed when explicitly requested. The precision of a power series is dynamically extended upon request, without requiring any re-computation of existing terms. This design extends into an “ancestry” of power series whereby power series created from the result of arithmetic or Weierstrass preparation automatically hold on to enough information to dynamically update themselves to higher precision using information from their “parents”.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
This library is accessible, yet undocumented, in Maple 2020 as RegularChains:-PowerSeries. See www.regularchains.org/documentation.html.
References
Alvandi, P., Ataei, M., Kazemi, M., Moreno Maza, M.: On the extended Hensel construction and its application to the computation of real limit points. J. Symb. Comput. 98, 120–162 (2020)
Alvandi, P., Kazemi, M., Moreno Maza, M.: Computing limits with the regularchains and powerseries libraries: from rational functions to zariski closure. ACM Commun. Comput. Algebra 50(3), 93–96 (2016)
Asadi, M., et al.: Basic Polynomial Algebra Subprograms (BPAS) (2020). http://www.bpaslib.org/
Asadi, M., Brandt, A., Moir, R.H.C., Moreno Maza, M., Xie, Y.: On the parallelization of triangular decomposition of polynomial systems. In: Proceedings of International Symposium on Symbolic and Algebraic Computation, ISSAC 2020, pp. 22–29. ACM (2020)
Asadi, M., Brandt, A., Moir, R.H.C., Moreno Maza, M.: Algorithms and data structures for sparse polynomial arithmetic. Mathematics 7(5), 441 (2019)
Burge, W.H., Watt, S.M.: Infinite structures in scratchpad II. In: Davenport, J.H. (ed.) EUROCAL 1987. LNCS, vol. 378, pp. 138–148. Springer, Heidelberg (1989). https://doi.org/10.1007/3-540-51517-8_103
Dahan, X., Moreno Maza, M., Schost, É., Wu, W., Xie, Y.: Lifting techniques for triangular decompositions. In: Proceedings of ISSAC 2005, Beijing, China, 2005, pp. 108–115 (2005)
Decker, W., Greuel, G.M., Pfister, G., Schönemann, H.: Singular 4-1-1 – a computer algebra system for polynomial computations (2018). http://www.singular.uni-kl.de
Fischer, G.: Plane Algebraic Curves. AMS (2001)
von zur Gathen, J., Gerhard, J.: Modern Computer Algebra, 2nd edn. Cambridge University Press, Cambridge (2003)
von zur Gathen, J.: Hensel and Newton methods in valuation rings. Math. Comput. 42(166), 637–661 (1984)
Hart, W., Johansson, F., Pancratz, S.: FLINT: Fast Library for Number Theory (2015), version 2.5.2. http://flintlib.org
van der Hoeven, J.: Relax, but don’t be too lazy. J. Symb. Comput. 34(6), 479–542 (2002)
Karczmarczuk, J.: Generating power of lazy semantics. Theor. Comput. Sci. 187(1–2), 203–219 (1997)
Kazemi, M., Moreno Maza, M.: Detecting singularities using the PowerSeries library. In: Gerhard, J., Kotsireas, I. (eds.) MC 2019. CCIS, vol. 1125, pp. 145–155. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-41258-6_11
Lauer, M.: Computing by homomorphic images. In: Buchberger, B., Collins, G.E., Loos, R., Albrecht, R. (eds.) Computer Algebra, pp. 139–168. Springer, Vienna (1983). https://doi.org/10.1007/978-3-7091-7551-4_10
McCool, M., Reinders, J., Robison, A.: Structured Parallel Programming: Patterns for Efficient Computation. Elsevier, Amsterdam (2012)
Monagan, M., Vrbik, P.: Lazy and forgetful polynomial arithmetic and applications. In: Gerdt, V.P., Mayr, E.W., Vorozhtsov, E.V. (eds.) CASC 2009. LNCS, vol. 5743, pp. 226–239. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-04103-7_20
Parusiński, A., Rond, G.: The Abhyankar-Jung theorem. J. Algebra 365, 29–41 (2012)
Sasaki, T., Kako, F.: Solving multivariate algebraic equation by Hensel construction. Japan J. Indust. Appl. Math. 16(2), 257–285 (1999)
Scott, M.L.: Programming Language Pragmatics, 3rd edn. Academic Press, Cambridge (2009)
The Sage Developers: SageMath, the Sage Mathematics Software System (Version 9.1) (2020). https://www.sagemath.org
Acknowledgments
The authors would like to thank NSERC of Canada (award CGSD3-535362-2019), Robert H. C. Moir, and the reviewers for their helpful comments.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2020 Springer Nature Switzerland AG
About this paper
Cite this paper
Brandt, A., Kazemi, M., Moreno-Maza, M. (2020). Power Series Arithmetic with the BPAS Library. In: Boulier, F., England, M., Sadykov, T.M., Vorozhtsov, E.V. (eds) Computer Algebra in Scientific Computing. CASC 2020. Lecture Notes in Computer Science(), vol 12291. Springer, Cham. https://doi.org/10.1007/978-3-030-60026-6_7
Download citation
DOI: https://doi.org/10.1007/978-3-030-60026-6_7
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-60025-9
Online ISBN: 978-3-030-60026-6
eBook Packages: Computer ScienceComputer Science (R0)