Gröbner bases and primary decomposition of polynomial ideals
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
Journal of Symbolic Computation
On lucky ideals for Gro¨bner basis computations
Journal of Symbolic Computation
Journal of Symbolic Computation
Efficient computation of zero-dimensional Gro¨bner bases by change of ordering
Journal of Symbolic Computation
Localization and primary decomposition of polynomial ideals
Journal of Symbolic Computation
Radical computations of zero-dimensional ideals and real root counting
Selected papers presented at the international IMACS symposium on Symbolic computation, new trends and developments
Computing the primary decomposition of zero-dimensional ideals
Journal of Symbolic Computation
An Algorithm for the Computation of the Radical of an Ideal in the Ring of Polynomials
AAECC-9 Proceedings of the 9th International Symposium, on Applied Algebra, Algebraic Algorithms and Error-Correcting Codes
ISAAC '88 Proceedings of the International Symposium ISSAC'88 on Symbolic and Algebraic Computation
Modular algorithms for computing Gröbner bases
Journal of Symbolic Computation
P-adic reconstruction of rational numbers
ACM SIGSAM Bulletin
Some comments on the modular approach to Gröbner-bases
ACM SIGSAM Bulletin
A Singular Introduction to Commutative Algebra
A Singular Introduction to Commutative Algebra
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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In this paper we investigate the parallelization of two modular algorithms. In fact, we consider the modular computation of Grobner bases (resp. standard bases) and the modular computation of the associated primes of a zero-dimensional ideal and describe their parallel implementation in Singular. Our modular algorithms for solving problems over Q mainly consist of three parts: solving the problem modulo p for several primes p, lifting the result to Q by applying the Chinese remainder algorithm (resp. rational reconstruction), and verification. Arnold proved using the Hilbert function that the verification part in the modular algorithm for computing Grobner bases can be simplified for homogeneous ideals (cf. Arnold, 2003). The idea of the proof could easily be adapted to the local case, i.e. for local orderings and not necessarily homogeneous ideals, using the Hilbert-Samuel function (cf. Pfister, 2007). In this paper we prove the corresponding theorem for non-homogeneous ideals in the case of a global ordering.