Numerical Decomposition of Affine Algebraic Varieties

An irreducible algebraic decomposition ∪i=0dXi=∪i=0d(∪j=1diXij)$ \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 a union of finite disjoint sets ∪i=0dWi=∪i=0d(∪j=1diWij)$\cup _{i = 0}^d W_i = \cup _{i = 0}^d ({\cup _{j = 1}^{d_i } W_{ij} } )$ called numerical irreducible decomposition (cf. [14],[15],[18],[19],[20],[22],[23],[24]). The W_i correspond to the pure i-dimensional components X_i, and the W_ij present the i-dimensional irreducible components X_ij. The numerical irreducible decomposition is implemented in Bertini (cf. [3]). The algorithms use homotopy continuation methods. We modify this concept using partially Gröbner bases, triangular sets, local dimension, and the so-called zero sum relation. We present in this paper the corresponding algorithms and their implementations in Singular (cf. [8]). We 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*.