Yet Another Ideal Decomposition Algorithm

The problem of decomposing an ideal into pure-dimensional components (resp. reduced pure-dimensional components) is a key step in several basic algorithms of commutative algebra. The computation of the radical can be performed as intersection of the reduced components (but we wonder why you should really perform this intersection that throws away additional insight in the structure of the ideal), and the primary or the irreducible reduced decomposition can be more easily obtained from a pure-dimensional decomposition. Moreover this step can be performed independently of the ground field —at least if it is perfect— while the primary and irreducible decompositions require factorization, that depends on the ground field. Several algorithms have been proposed for this computation, and fall mainly into two classes: the family of projection algorithms, whose prototype is the primary decomposition algorithm of [15], and the direct, syzygy-based algorithms, like [11], see also [23]. The superiority of one type of algorithms over the other has not been settled; in [11] it is argued that the direct methods are superior since for projection methods “sufficiently generic” projections are needed. This assertion is only marginally true for the current literature — already in [3] “generic linear combinations” are needed only for finding a single univariate polynomial, through linear algebra and a variation of [13], and the algorithms of [19] and [17] compute triangular decompositions through characteristic sets without changes of coordinates. Moreover projection algorithms do not have the limitation to large characteristics of [11], in particular the example of D. Jaffe in [11] can be easily handled by our algorithm (and indeed by all projection algorithms). In this paper we describe algorithms for equidimensional decompositions, that can be seen as a marginal modification of an algorithm of [3], but that completely avoid generic or random projections, and does not need lexicographic Gröbner bases or characteristic sets, hence can be a candidate to a best competitor against direct algorithms. Some tests seem to support the belief that our algorithms are much faster than direct algorithms.