Numerical Decomposition of Affine Algebraic Varieties

zbMATH Open1469.14115arXiv1010.3129MaRDI QIDQ2937222

Publication date: 8 January 2015

Abstract: An irreducible algebraic decomposition

c u p_{i = 0}^{d} X_{i} = c u p_{i = 0}^{d} (c u p_{j = 1}^{d_{i}} X_{i j})

of an affine algebraic variety X can be represented as an union of finite disjoint sets

c u p_{i = 0}^{d} W_{i} = c u p_{i = 0}^{d} (c u p_{j = 1}^{d_{i}} W_{i j})

called numerical irreducible decomposition (cf. [14],[15],[17],[18],[19],[21],[22],[23]).

W_{i}

corresponds to a pure i-dimensional

X_{i}

, and

W_{i j}

presents an i- dimensional irreducible component

X_{i j}

. Modifying this concepts by using partially Gr"obner bases, local dimension, and the "Zero Sum Relation" we present in this paper an implementation in SINGULAR to compute the numerical irreducible decomposition. We will give some examples and timings, which show that the modified algorithms are more efficient if the number of variables is not too large. For a large number of variables BERTINI is more efficient. Note that each step of the numerical decomposition is parallelizable. For our comparisons we did not use the parallel version of BERTINI.

Full work available at URL: https://arxiv.org/abs/1010.3129

zbMATH Keywords

local dimension monodromy action Gr\obner basishomotopy function witness point set zero sum relation

Mathematics Subject Classification ID

Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) (13P10) Computational aspects of higher-dimensional varieties (14Q15)