Exact decomposition of algebraic varieties using numerical techniques

Exact techniques (i.e, Gröbner bases) for solving systems of polynomial equations become
computationally infeasible for algebraic varieties defined by a large number of equations. In this lecture, I will describe an effective technique for computing irreducible decompositions of varieties defined by ~ 1,000,000 polynomials in ~ 100 dimensional space. The algorithm first uses a numerical optimization technique to compute witness points, which are floating point approximations to exact solutions of the system. Lattice reduction is then used to jump from a suitably accurate witness point to a set of exact polynomials that describe
an irreducible component of the variety. An open source implementation of the algorithm using the Sage computer algebra system is being developed, optimized for parallel computation using dozens of processors. Applications in quantum mechanics will be discussed.