Numerical Decomposition of Affine Algebraic Varieties

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 a 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],[18],[19],[20],[22],[23],[24]). The Wi correspond to the pure i-dimensional components Xi, and the Wij present the i-dimensional irreducible components Xij. 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*.

Computing Singular Points of Projective Plane Algebraic Curves by Homotopy Continuation Methods

We present an algorithm that computes the singular points of projective plane algebraic curves and determines their multiplicities and characters. The feasibility of the algorithm is analyzed. We prove that the algorithm has the polynomial time complexity on the degree of the algebraic curve. The algorithm involves the combined applications of homotopy continuation methods and a method of root computation of univariate polynomials. Numerical experiments show that our algorithm is feasible and efficient.