The use of bad primes in rational reconstruction
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- by Janko Böhm, Wolfram Decker, Claus Fieker and Gerhard Pfister PDF
- Math. Comp. 84 (2015), 3013-3027 Request permission
Abstract:
A standard method for finding a rational number from its values modulo a collection of primes is to determine its value modulo the product of the primes via Chinese remaindering, and then use Farey sequences for rational reconstruction. Successively enlarging the set of primes if needed, this method is guaranteed to work if we restrict ourselves to “good” primes. Depending on the particular application, however, there may be no efficient way of identifying good primes.
In the algebraic and geometric applications we have in mind, the final result consists of an a priori unknown ideal (or module) which is found via a construction yielding the (reduced) Gröbner basis of the ideal. In this context, we discuss a general setup for modular and, thus, potentially parallel algorithms which can handle “bad” primes. A new key ingredient is an error tolerant algorithm for rational reconstruction via Gaussian reduction.
References
- William W. Adams and Philippe Loustaunau, An introduction to Gröbner bases, Graduate Studies in Mathematics, vol. 3, American Mathematical Society, Providence, RI, 1994. MR 1287608, DOI 10.1090/gsm/003
- Elizabeth A. Arnold, Modular algorithms for computing Gröbner bases, J. Symbolic Comput. 35 (2003), no. 4, 403–419. MR 1976575, DOI 10.1016/S0747-7171(02)00140-2
- Janko Böhm, Wolfram Decker, Santiago Laplagne, Gerhard Pfister, Andreas Steenpaß, and Stefan Steidel, Parallel algorithms for normalization, J. Symbolic Comput. 51 (2013), 99–114. MR 3005784, DOI 10.1016/j.jsc.2012.07.002
- J. Böhm, W. Decker, S. Laplagne, and G. Pfister, Local to global algorithms for the Gorenstein adjoint ideal of a curve. In preparation.
- George E. Collins and Mark J. Encarnación, Efficient rational number reconstruction, J. Symbolic Comput. 20 (1995), no. 3, 287–297. MR 1378101, DOI 10.1006/jsco.1995.1051
- Mark J. Encarnación, Computing GCDs of polynomials over algebraic number fields, J. Symbolic Comput. 20 (1995), no. 3, 299–313. MR 1378102, DOI 10.1006/jsco.1995.1052
- Gert-Martin Greuel, Santiago Laplagne, and Frank Seelisch, Normalization of rings, J. Symbolic Comput. 45 (2010), no. 9, 887–901. MR 2661161, DOI 10.1016/j.jsc.2010.04.002
- Nazeran Idrees, Gerhard Pfister, and Stefan Steidel, Parallelization of modular algorithms, J. Symbolic Comput. 46 (2011), no. 6, 672–684. MR 2781946, DOI 10.1016/j.jsc.2011.01.003
- Phong Q. Nguyen and Damien Stehlé, Low-dimensional lattice basis reduction revisited, ACM Trans. Algorithms 5 (2009), no. 4, Art. 46, 48. MR 2571909, DOI 10.1145/1597036.1597050
- Peter Kornerup and R. T. Gregory, Mapping integers and Hensel codes onto Farey fractions, BIT 23 (1983), no. 1, 9–20. MR 689601, DOI 10.1007/BF01937322
- P. S. Wang, A p–adic algorithm for univariate partial fractions. Proceedings SYMSAC ’81, 212–217 (1981).
- P. S. Wang, M. J. T. Guy, and J. H. Davenport, P–adic reconstruction of rational numbers. SIGSAM Bull, 2–3 (1982).
Additional Information
- Janko Böhm
- Affiliation: Fachbereich Mathematik, Technical University Kaiserslautern, Postfach 3049, 67653 Kaiserslautern, Germany
- MR Author ID: 974387
- ORCID: 0000-0003-1702-5864
- Email: boehm@mathematik.uni-kl.de
- Wolfram Decker
- Affiliation: Fachbereich Mathematik, Technical University Kaiserslautern, Postfach 3049, 67653 Kaiserslautern, Germany
- Email: decker@mathematik.uni-kl.de
- Claus Fieker
- Affiliation: Fachbereich Mathematik, Technical University Kaiserslautern, Postfach 3049, 67653 Kaiserslautern, Germany
- Email: fieker@mathematik.uni-kl.de
- Gerhard Pfister
- Affiliation: Fachbereich Mathematik, Technical University Kaiserslautern, Postfach 3049, 67653 Kaiserslautern, Germany
- Email: pfister@mathematik.uni-kl.de
- Received by editor(s): July 16, 2012
- Received by editor(s) in revised form: September 27, 2013, and March 24, 2014
- Published electronically: April 15, 2015
- © Copyright 2015 American Mathematical Society
- Journal: Math. Comp. 84 (2015), 3013-3027
- MSC (2010): Primary 13P10, 68W10; Secondary 52C05
- DOI: https://doi.org/10.1090/mcom/2951
- MathSciNet review: 3378860