Parallelization of Modular Algorithms

Details Parallelization of Modular Algorithms Parallelization of Modular Algorithms

2010-05-31

arxiv.org/abs/1005.5663v2

Journal of Symbolic Computation 46, 672-684 (2011)

Mathematics

Commutative Algebra

16 pages

Scientific paper

10.1016/j.jsc.2011.01.003

In this paper we investigate the parallelization of two modular algorithms. In fact, we consider the modular computation of Gr\"obner 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 to solve problems over Q mainly consist of three parts, solving the problem modulo p for several primes p, lifting the result to Q by applying Chinese remainder resp. rational reconstruction, and a part of verification. Arnold proved using the Hilbert function that the verification part in the modular algorithm to compute Gr\"obner bases can be simplified for homogeneous ideals (cf. \cite{A03}). 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. \cite{Pf07}). In this paper we prove the corresponding theorem for non-homogeneous ideals in case of a global ordering.

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