The use of bad primes in rational reconstruction

Böhm, Janko; Decker, Wolfram; Fieker, Claus; Pfister, Gerhard

doi:10.1090/mcom/2951

The use of bad primes in rational reconstruction
HTML articles powered by AMS MathViewer

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).