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.Comment: 16 page

In this article we present two new algorithms to compute the Grobner basis of an ideal that is invariant under certain permutations of the ring variables and which are both implemented in SINGULAR (cf. Decker et al., 2012). The first and major algorithm is most performant over finite fields whereas the second algorithm is a probabilistic modification of the modular computation of Grobner bases based on the articles by Arnold (cf. Arnold, 2003), Idrees, Pfister, Steidel (cf. Idrees et al., 2011) and Noro, Yokoyama (cf. Noro and Yokoyama, in preparation; Yokoyama, 2012). In fact, the first algorithm that mainly uses the given symmetry, improves the necessary modular calculations in positive characteristic in the second algorithm. Particularly, we could, for the first time even though probabilistic, compute the Grobner basis of the famous ideal of cyclic 9-roots (cf. Bjorck and Froberg, 1991) over the rationals with SINGULAR

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

No abstract

Modern vehicles are highly complex systems consisting of many subsystems in various physical domains, e.g. electric, electronic, hydraulic and control systems that dynamically interact. For any subsystem, there are tailored simulation tools with specifically developed and adapted numerical solvers. In this context, co-simulation strategies are particularly attractive, where each submodel is solved with appropriate numerical methods. In industrial applications, one is hereby confronted with enormous numerical challenges with respect to efficiency, accuracy and numerical stability -especially in online applications. In this article, we present co-simulation strategies by means of selected application examples from the field of vehicle engineering.