Parallel algorithms for normalization

TL;DR

This paper introduces parallel algorithms for computing the normalization of affine algebras, improving efficiency by stratifying the singular locus and employing modular computations, with implementations showing significant performance gains.

Contribution

It presents novel parallel and modular algorithms for normalization, enhancing existing methods and demonstrating improved performance in computer algebra systems.

Findings

Algorithms outperform previous methods in most examples

Parallel and modular approaches improve efficiency

Implementation in SINGULAR confirms performance gains

Abstract

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.