Triangular systems and factorized Gröbner bases

Abstract

Solving systems of polynomial equations in an ultimate way means to nd the isolated primes of the associated variety and to present them in a way that is well suited for further computations.[5] proposes an algorithm, that uses several Grobner basis computations for a dimension reduction argument, delaying factorization to the end of the algorithm. In this paper we i n v estigate the opposite approach, heavily using factorization (of multivariate polynomials), delaying the computation of stable ideal quotients. At a heuristic level this is exactly the well known Grobner algorithm with factorization and constraint inequalities, available in all major computer algebra systems. In a preceding paper [9] we reported on some experience with a new version of this algorithm, implemented in our REDUCE package CALI [8]. Here we discuss, how this approach may bere ned to produce triangular systems in the sense of [12] and [13]. Such a re nement guarantees, di erent to the usual Grobner factorizer, to produce a quasi prime decomposition, i.e. the resulting components are at least pure dimensional radical ideals. As in [9] our method weakens the usual restriction to lexicographic term orders.

Minimal Primary Decomposition and Factorized Gröbner Bases

“…This paper continues our study of applications of factorized Grobner basis computations in [8] and [9].…”

“…It turns out empirically, that the same advantage, observed for the factorized Grobner basis algorithm in contrast to the ordinary one solving polynomial systems of equations in [8] and [9] for well splitting examples, holds also for the computation of isolated primes. Of course, this re ects the general observation, that usually geometric properties of ideals (here: the computation of isolated primes) are computationally more handy than algebraic ones (here: the computation of primary decompositions).…”

“…It invokes factorization of reduced S-polynomials during the calculation of Grobner bases and splits the computation into as many branches as (di erent) factors occur. Since the algorithm is part of almost all general purpose Computer Algebra Systems, we will not describe it here and refer the reader to [8] and [9] instead, where we discussed this algorithm in great detail and employed it successfully to decompose a given set of polynomials into triangular systems.…”

“…F or this purpose de ne a As explained e.g. in [9] one can compute denominator-free inF using the well known pseudo normal form algorithm PNF p,B . F or p 2 F it returns a denominator-free pseudoS-normal form p 0 2 F F with respect to B, 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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