Polyhedral Entropy and Random Polynomial Systems
Abstract: We present a conjecture relating random polynomial equations, polytopes, and Kahler geometry. Roughly speaking, the Square Root Volume Conjecture (SRVC), is a generalization to sparse polynomial systems of the fact that the expected number of real roots of a random dense polynomial system is the square root of the expected number of complex roots. The latter result is due to Shub and Smale, and has since been extended in various directions. However, little work has been on the polytopal/sparse polynomial systems side.
We review some recent progress on the conjecture, as well as the origins of the square root phenomenon. Along the way, some hard integrals and simple lattice point constructions will appear. We then present some applications of our framework to the numerics of polynomial system solving and Smale's 17th Problem.