  Consider the set of solutions to a system of polynomial equations in many
variables. An algebraic manifold is an open submanifold of such a set. We
introduce a new method for computing integrals and sampling from distributions
on algebraic manifolds. This method is based on intersecting with random linear
spaces. It produces i.i.d. samples, works in the presence of multiple connected
components, and is simple to implement. We present applications to
computational statistical physics and topological data analysis.
