Parallelization of Modular Algorithms

TL;DR

This paper explores the parallelization of modular algorithms for computing Gr"obner bases and associated primes, implementing and analyzing their efficiency and correctness in the SINGULAR system.

Contribution

It introduces parallel implementations of modular algorithms for Gr"obner bases and associated primes, extending verification techniques to non-homogeneous ideals with global orderings.

Findings

Parallel algorithms improve computational efficiency.

Verification methods are adapted for non-homogeneous ideals.

Implementation in SINGULAR demonstrates practical applicability.

Abstract

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.