Numerically intersecting algebraic varieties via witness sets

The fundamental construct of numerical algebraic geometry is the representation of an irreducible algebraic set, A, by a witness set, which consists of a polynomial system, F, for which A   is an irreducible component of V(F)V(F), a generic linear space LL of complementary dimension to A  , and a numerical approximation to the set of witness points, L∩AL∩A. Given F  , methods exist for computing a numerical irreducible decomposition, which consists of a collection of witness sets, one for each irreducible component of V(F)V(F). This paper concerns the more refined question of finding a numerical irreducible decomposition of the intersection A∩BA∩B of two irreducible algebraic sets, A and B, given a witness set for each. An existing algorithm, the diagonal homotopy  , computes witness point supersets for A∩BA∩B, but this does not complete the numerical irreducible decomposition. In this paper, we use the theory of isosingular sets to complete the process of computing the numerical irreducible decomposition of the intersection.