Localization and Primary Decomposition of Polynomial Ideals

Abstract

In this paper, we propose a new method for primary decomposition of a polynomial ideal, not necessarily zero-dimensional, and report on a detailed study for its practical implementation. In our method, we introduce two key techniques, effective localization and fast elimination of redundant components, by which we can get a good performance for several examples. The performance of our method is examined by comparison with other existing methods based on practical experiments.

Minimal Primary Decomposition and Factorized Gröbner Bases

“…Based on that observation we generalize the algorithm presented in [22] t o t h e computation of primary decompositions for modules. It rests on an ideal separation argument.…”

“…Regardless of the wide attention that this theoretical work attracted in the community, up to now there are only a few implementations of the algorithm: As far as we know, the AXIOM implementation of the authors of [5], an implementation in MAS by H. Kredel for zero dimensional ideals, our implementation in the REDUCE package CALI [7] and the implementation in the computer algebra system Risa/Asir [15] b y the authors of [22]. Only CALI o ers primary decomposition also for modules.…”

“…During the preparation of this paper we became aware of analogous considerations in [22], allowing several shortcuts compared to an earlier version of this paper. Di erent from [22], where the authors consider only primary decomposition of ideals, results are explained here in general for pairs of submodules N M of a nitely generated free module F.…”

“…To extract nally the primary components we use as in [22] ideal separators with respect to a list of isolated primes, computed in advance. We generalize this approach to a (relative) module situation, too, i.e.…”

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AbstractThis paper continues our study of applications of factorized Grobner basis computations in [8] and [9].We describe a way t o i n terweave factorized Grobner bases and the ideas in [5] that leads to a signi cant speed up in the computation of isolated primes for well splitting examples.Based on that observation we generalize the algorithm presented in [22] t o t h e computation of primary decompositions for modules. It rests on an ideal separation argument.

We also discuss the practically important question how to extract a minimal primary decomposition, neither addressed in [5] nor in [17].

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