Segre-Driven Containment Testing for Varieties

In this talk we consider new methods to test pairwise containment of projective vareities (i.e. the solutions to systems of homogenous polynomial equations). Previous aproaches to this problem have doubly exponetial complexity, in this talk we present a new effective algorithm with singly exponetial complexity. More generally this algorithm can test pairwise containment of arbitrary (possibly singular) subvarieties of any smooth projective toric variety and can also determine algebraic multiplicity without working in local rings. These methods may be implemented without using Gröbner bases; in particular numeric methods can be used. The methods arise from techniques developed to compute the Segre class $s(X,Y)$ of $X$ in $Y$ for $X$ and $Y$ arbitrary subschemes of some smooth projective toric variety $T$. In particular, this work also gives an explicit method to compute these Segre classes and other associated objects such as the Fulton-MacPherson intersection product of projective varieties (this intersection product captures the behavior of intersections of two varieites inside a third). These algorithms are implemented in Macaulay2 and have been found to be effective on a variety of examples. This is joint work with Corey Harris (University of Oslo).