Numerical algebraic intersection using regeneration

In numerical algebraic geometry, algebraic sets are represented by witness sets. This paper presents an algorithm, based on the regeneration technique, that solves the following problem: given a witness set for a pure-dimensional algebraic set Z, along with a system of polynomial equations f : Z Ñ C, compute a numerical irreducible decomposition of V Z X Vpfq. An important special case is when Z A B for irreducible sets A and B and fpx, yq x y for x P A, y P B, in which case V is isomorphic to A X B. In this way, the current contribution is a generalization of existing diagonal intersection techniques. Another important special case is when Z A C, so that the projection of V dropping the last k coordinates consists of the points x P A where there exists some y in a new set of variables such that fpx, yq 0. This arises in many contexts, such as finding the singularities of A, in which case fpx, yq can be a set of singularity conditions that involve new variables associated to the tangent space of A. The combining of multiple intersection scenarios into one common scheme brings new capabilities and organizational simplification to numerical algebraic geometry.