Parallel algorithms for normalization

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.Comment: 19 page

Desingularization of Arithmetic Surfaces: Algorithmic Aspects

“…Although there has been significant improvement in the efficiency of Grauert-Remmert style normalization algorithms in the last decade (see e.g. [14], [1]), this is still a bottleneck when working over a Dedekind domain R instead of a field. The crucial step here is the choice of a suitable test ideal, i.e.…”

Current Challenges in Developing Open Source Computer Algebra Systems

“…The systematic design of parallel algorithms for applications which so far can only be handled by sequential algorithms is a major task for the years to come. For normalization, this problem has recently been solved [14]. Over the field of rational numbers, the new algorithm becomes particularly powerful by combining it with modular methods, see again Section 3.…”

“…As already pointed out, it is a major challenge to systematically design parallel alternatives to basic and high-level algorithms which are sequential in nature. For normalization, this problem has recently been solved in [14] by using the technique of localization and proving a local version of the Grauert-Remmert normality criterion.…”

“…Composing this with the projection, we get the normalization map from Example 10. Now, following [14], we describe how the normalization algorithm can be redesigned so that it becomes parallel in nature. For simplicity of the presentation, we focus on the case where Sing(A) is a finite set.…”

Bad Primes in Computational Algebraic Geometry

“…The Gorenstein adjoint ideal can be computed via a local-to-global strategy. The minimal local contribution to A at P is unique and can be computed using Grauert-Remmert-type normalization algorithms, see [4]. It can be written as B = U d with an ideal U ⊂ A and a common denominator d ∈ A.…”