Parallel algorithms for normalization

Abstract

Given a reduced affine algebra A over a perfect field K, we present parallel algorithms to compute the normalization $\overline{A}$ of A. Our starting point is the algorithm of Greuel et al. (2010), which is an improvement of de Jongʼs algorithm (de Jong, 1998, Decker et al., 1999). First, we propose to stratify the singular locus $\mathrm{Sing}(A)$ in a way which is compatible with normalization, apply a local version of the normalization algorithm at each stratum, and find $\overline{A}$ by putting the local results together. Second, in the case where $K=\mathbb{Q}$ 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 et al. (2010). In the case where $K=\mathbb{Q}$, we also discuss the use of modular computations of Gröbner bases, radicals, and primary decompositions. We point out that in most examples, the new algorithms outperform the algorithm of Greuel et al. (2010) by far, even if we do not run them in parallel.