Computer algebra: systems and algorithms for algebraic computation
Computer algebra: systems and algorithms for algebraic computation
On an installation of Buchberger's algorithm
Journal of Symbolic Computation
A p-adic approach to the computation ofGröbner bases
Journal of Symbolic Computation
A modular method for Gro¨bner-basis construction over Q and solving system of algebraic equations
Journal of Information Processing
On lucky ideals for Gro¨bner basis computations
Journal of Symbolic Computation
Journal of Symbolic Computation
Hilbert functions and the Buchberger algorithm
Journal of Symbolic Computation
ISAAC '88 Proceedings of the International Symposium ISSAC'88 on Symbolic and Algebraic Computation
A criterion for detecting unnecessary reductions in the construction of Groebner bases
EUROSAM '79 Proceedings of the International Symposiumon on Symbolic and Algebraic Computation
Upper and Lower Bounds for the Degree of Groebner Bases
EUROSAM '84 Proceedings of the International Symposium on Symbolic and Algebraic Computation
Computing grobner bases with hilbert lucky primes
Computing grobner bases with hilbert lucky primes
Some comments on the modular approach to Gröbner-bases
ACM SIGSAM Bulletin
Maximal quotient rational reconstruction: an almost optimal algorithm for rational reconstruction
ISSAC '04 Proceedings of the 2004 international symposium on Symbolic and algebraic computation
Lifting techniques for triangular decompositions
Proceedings of the 2005 international symposium on Symbolic and algebraic computation
Proceedings of the 2006 international symposium on Symbolic and algebraic computation
Computing Gröbner bases of ideals of few points in high dimensions
ACM Communications in Computer Algebra
Gröbner bases for families of affine or projective schemes
Journal of Symbolic Computation
Efficient Algorithms for Computing Nœther Normalization
Computer Mathematics
Computing Equiangular Lines in Complex Space
Mathematical Methods in Computer Science
Proceedings of the 2009 international symposium on Symbolic and algebraic computation
Modular Algorithms for Computing a Generating Set of the Syzygy Module
CASC '09 Proceedings of the 11th International Workshop on Computer Algebra in Scientific Computing
Parallelization of Modular Algorithms
Journal of Symbolic Computation
Programming and Computing Software
Usage of modular techniques for efficient computation of ideal operations
CASC'12 Proceedings of the 14th international conference on Computer Algebra in Scientific Computing
Parallel algorithms for normalization
Journal of Symbolic Computation
Gröbner bases of symmetric ideals
Journal of Symbolic Computation
Parallel modular computation of Gröbner and involutive bases
Programming and Computing Software
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Intermediate coefficient swell is a well-known difficulty with Buchberger's algorithm for computing Gröbner bases over the rational numbers. p-Adic and modular methods have been successful in limiting intermediate coefficient growth in other computations, and in particular in the Euclidian algorithm for computing the greatest common divisor (GCD) of polynomials in one variable. In this paper we present two modular algorithms for computing a Gröbner basis for an ideal in Q[x1,..., xv] which extend the modular GCD algorithms. These algorithms improve upon previously proposed modular techniques for computing Gröbner bases in that we test primes before lifting, and also provide an algorithm for checking the result for correctness. A complete characterization of unlucky primes is also given. Finally, we give some preliminary timings which indicate that these modular algorithms can provide considerable time improvements in examples where intermediate coefficient growth is a problem.