Numerical Decomposition of Affine Algebraic Varieties

TL;DR

This paper introduces an improved algorithm for numerical irreducible decomposition of affine algebraic varieties using Gr"obner bases and the Zero Sum Relation, implemented in SINGULAR, with efficiency advantages for moderate variable counts.

Contribution

The paper presents a novel implementation of numerical irreducible decomposition leveraging partial Gr"obner bases and local dimension, enhancing efficiency for problems with fewer variables.

Findings

Modified algorithms outperform existing methods for moderate variable counts.

Implementation in SINGULAR demonstrates practical efficiency improvements.

Parallelizable steps offer potential for further speedups.

Abstract

An irreducible algebraic decomposition $\cup_{i=0}^{d}X_i=\cup_{i=0}^{d} (\cup_{j=1}^{d_i}X_{ij})$ of an affine algebraic variety X can be represented as an union of finite disjoint sets $\cup_{i=0}^{d}W_i=\cup_{i=0} ^{d}(\cup_{j=1}^{d_i}W_{ij})$ called numerical irreducible decomposition (cf. [14],[15],[17],[18],[19],[21],[22],[23]). $W_i$ corresponds to a pure i-dimensional $X_i$, and $W_{ij}$ presents an i- dimensional irreducible component $X_{ij}$. 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.