Abstract
Let \(\mathcal {C}\) be a plane curve given by an equation \(f(x,y)=0\) with \(f\in K[x][y]\) a monic irreducible polynomial. We study the problem of computing an integral basis of the algebraic function field \(K(\mathcal {C})\) and give new complexity bounds for three known algorithms dealing with this problem. For each algorithm, we study its subroutines and, when it is possible, we modify or replace them so as to take advantage of faster primitives. Then, we combine complexity results to derive an overall complexity estimate for each algorithm. In particular, we modify an algorithm due to Böhm et al. and achieve a quasi-optimal runtime.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Arvind, V., Vijayaraghavan, T.C.: The complexity of solving linear equations over a finite ring. In: Diekert, V., Durand, B. (eds.) STACS 2005. LNCS, vol. 3404, pp. 472–484. Springer, Heidelberg (2005). https://doi.org/10.1007/978-3-540-31856-9_39
Bauch, J.D.: Computation of integral bases. J. Number Theory 165, 382–407 (2016)
de Beaudrap, N.: On the complexity of solving linear congruences and computing nullspaces modulo a constant. arXiv preprint arXiv:1202.3949 (2012)
Böhm, J., Decker, W., Laplagne, S., Pfister, G.: Computing integral bases via localization and Hensel lifting. arXiv preprint arXiv:1505.05054 (2015)
Böhm, J., Decker, W., Laplagne, S., Pfister, G., Steenpaß, A., Steidel, S.: Parallel algorithms for normalization. J. Symb. Comput. 51, 99–114 (2013)
Bostan, A., et al.: Algorithmes efficaces en calcul formel (2017)
De Jong, T., Pfister, G.: Local Analytic Geometry: Basic Theory and Applications. Springer, Wiesbaden (2013). https://doi.org/10.1007/978-3-322-90159-0
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
Diem, C.: On arithmetic and the discrete logarithm problem in class groups of curves. Habilitation, Universität Leipzig (2009)
Duval, D.: Rational Puiseux expansions. Compositio Mathematica 70(2), 119–154 (1989)
Hess, F.: Computing Riemann-Roch spaces in algebraic function fields and related topics. J. Symb. Comput. 33(4), 425–445 (2002)
van Hoeij, M.: An algorithm for computing an integral basis in an algebraic function field. J. Symb. Comput. 18(4), 353–363 (1994)
van Hoeij, M., Novocin, A.: A reduction algorithm for algebraic function fields (2008)
van Hoeij, M., Stillman, M.: Computing an integral basis for an algebraic function field (2015). https://www.math.fsu.edu/~hoeij/papers/2015/slides.pdf
van der Hoeven, J., Lecerf, G.: Directed evaluation, December 2018. Working paper or preprint. https://hal.archives-ouvertes.fr/hal-01966428
Kedlaya, K.S., Umans, C.: Fast polynomial factorization and modular composition. SIAM J. Comput. 40(6), 1767–1802 (2011)
Labahn, G., Neiger, V., Zhou, W.: Fast, deterministic computation of the Hermite normal form and determinant of a polynomial matrix. J. Complexity 42, 44–71 (2017)
Le Gall, F.: Powers of tensors and fast matrix multiplication. In: Proceedings of the 39th International Symposium on Symbolic and Algebraic Computation, pp. 296–303 (2014)
Moroz, G., Schost, É.: A fast algorithm for computing the truncated resultant. In: Proceedings of the ACM on International Symposium on Symbolic and Algebraic Computation, pp. 341–348 (2016)
Neiger, V.: Fast computation of shifted Popov forms of polynomial matrices via systems of modular polynomial equations. In: Proceedings of the ACM on International Symposium on Symbolic and Algebraic Computation, pp. 365–372 (2016)
Poteaux, A., Weimann, M.: Computing Puiseux series: a fast divide and conquer algorithm. arXiv preprint arXiv:1708.09067 (2017)
Trager, B.M.: Algorithms for manipulating algebraic functions. SM thesis MIT (1976)
Trager, B.M.: Integration of algebraic functions. Ph.D. thesis, Massachusetts Institute of Technology (1984)
Walker, R.J.: Algebraic curves (1950)
Zassenhaus, H.: Ein algorithmus zur berechnung einer minimalbasis über gegebener ordnung. In: Collatz, L., Meinardus, G., Unger, H. (eds.) Funktionalanalysis Approximationstheorie Numerische Mathematik, pp. 90–103. Springer, Basel (1967). https://doi.org/10.1007/978-3-0348-5821-2_10
Acknowledgments
Part of this work was completed while the author was at the Symbolic Computation Group of the University of Waterloo. This paper is part of a project that has received funding from the French Agence de l’Innovation de Défense. The author is grateful to Grégoire Lecerf, Adrien Poteaux and Éric Schost for helpful discussions and to Grégoire Lecerf for feedback on a preliminary version of this paper. The author also wishes to thank the anonymous reviewers for their 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
Abelard, S. (2020). On the Complexity of Computing Integral Bases of Function Fields. 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_3
Download citation
DOI: https://doi.org/10.1007/978-3-030-60026-6_3
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)