Parallelization of Modular Algorithms

In this paper we investigate the parallelization of two modular algorithms. In fact, we consider the modular computation of Gröbner 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 for solving problems over Q mainly consist of three parts: solving the problem modulo p for several primes p, lifting the result to Q by applying the Chinese remainder algorithm (resp. rational reconstruction), and verification. Arnold proved using the Hilbert function that the verification part in the modular algorithm for computing Gröbner bases can be simplified for homogeneous ideals (cf. Arnold, 2003, ). 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. Pfister, 2007). In this paper we prove the corresponding theorem for non-homogeneous ideals in the case of a global ordering.