2,677 research outputs found
Resolving zero-divisors using Hensel lifting
Algorithms which compute modulo triangular sets must respect the presence of
zero-divisors. We present Hensel lifting as a tool for dealing with them. We
give an application: a modular algorithm for computing GCDs of univariate
polynomials with coefficients modulo a radical triangular set over the
rationals. Our modular algorithm naturally generalizes previous work from
algebraic number theory. We have implemented our algorithm using Maple's RECDEN
package. We compare our implementation with the procedure RegularGcd in the
RegularChains package.Comment: Shorter version to appear in Proceedings of SYNASC 201
Reconstructing Rational Functions with
We present the open-source library for the
reconstruction of multivariate rational functions over finite fields. We
discuss the involved algorithms and their implementation. As an application, we
use in the context of integration-by-parts reductions and
compare runtime and memory consumption to a fully algebraic approach with the
program .Comment: 46 pages, 3 figures, 6 tables; v2: matches published versio
Gr\"obner Bases over Algebraic Number Fields
Although Buchberger's algorithm, in theory, allows us to compute Gr\"obner
bases over any field, in practice, however, the computational efficiency
depends on the arithmetic of the ground field. Consider a field , a simple extension of , where is an
algebraic number, and let be the minimal polynomial of
. In this paper we present a new efficient method to compute Gr\"obner
bases in polynomial rings over the algebraic number field . Starting from
the ideas of Noro [Noro, 2006], we proceed by joining to the ideal to be
considered, adding as an extra variable. But instead of avoiding
superfluous S-pair reductions by inverting algebraic numbers, we achieve the
same goal by applying modular methods as in [Arnold, 2003; B\"ohm et al., 2015;
Idrees et al., 2011], that is, by inferring information in characteristic zero
from information in characteristic . For suitable primes , the
minimal polynomial is reducible over . This allows us to
apply modular methods once again, on a second level, with respect to the
factors of . The algorithm thus resembles a divide and conquer strategy and
is in particular easily parallelizable. At current state, the algorithm is
probabilistic in the sense that, as for other modular Gr\"obner basis
computations, an effective final verification test is only known for
homogeneous ideals or for local monomial orderings. The presented timings show
that for most examples, our algorithm, which has been implemented in SINGULAR,
outperforms other known methods by far.Comment: 16 pages, 1 figure, 1 tabl
Development of symbolic algorithms for certain algebraic processes
This study investigates the problem of computing the exact greatest common divisor of two polynomials relative to an orthogonal basis, defined over the rational number field. The main objective of the study is to design and implement an effective and efficient symbolic algorithm for the general class of dense polynomials, given the rational number defining terms of their basis. From a general algorithm using the comrade matrix approach, the nonmodular and modular techniques are prescribed. If the coefficients of the generalized polynomials are multiprecision integers, multiprecision arithmetic will be required in the construction of the comrade matrix and the corresponding systems coefficient matrix. In addition, the application of the nonmodular elimination technique on this coefficient matrix extensively applies multiprecision rational number operations. The modular technique is employed to minimize the complexity involved in such computations. A divisor test algorithm that enables the detection of an unlucky reduction is a crucial device for an effective implementation of the modular technique. With the bound of the true solution not known a priori, the test is devised and carefully incorporated into the modular algorithm. The results illustrate that the modular algorithm illustrate its best performance for the class of relatively prime polynomials. The empirical computing time results show that the modular algorithm is markedly superior to the nonmodular algorithms in the case of sufficiently dense Legendre basis polynomials with a small GCD solution. In the case of dense Legendre basis polynomials with a big GCD solution, the modular algorithm is significantly superior to the nonmodular algorithms in higher degree polynomials. For more definitive conclusions, the computing time functions of the algorithms that are presented in this report have been worked out. Further investigations have also been suggested
Parallel algorithms for normalization
Given a reduced affine algebra A over a perfect field K, we present parallel
algorithms to compute the normalization \bar{A} of A. Our starting point is the
algorithm of Greuel, Laplagne, and Seelisch, which is an improvement of de
Jong's algorithm. First, we propose to stratify the singular locus Sing(A) in a
way which is compatible with normalization, apply a local version of the
normalization algorithm at each stratum, and find \bar{A} by putting the local
results together. Second, in the case where K = Q is the field of rationals, we
propose modular versions of the global and local-to-global algorithms. We have
implemented our algorithms in the computer algebra system SINGULAR and compare
their performance with that of the algorithm of Greuel, Laplagne, and Seelisch.
In the case where K = Q, we also discuss the use of modular computations of
Groebner bases, radicals, and primary decompositions. We point out that in most
examples, the new algorithms outperform the algorithm of Greuel, Laplagne, and
Seelisch by far, even if we do not run them in parallel.Comment: 19 page
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