Parallel Algorithms for Normalization

Details Parallel Algorithms for Normalization Parallel Algorithms for Normalization

2011-10-19

arxiv.org/abs/1110.4299v1

Mathematics

Commutative Algebra

15 pages

Scientific paper

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 algorithms. We have implemented our algorithms in the computer algebra system SINGULAR and compare their performance with that of other algorithms. 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.

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