Why Polyhedra Matter in Non-Linear Equation Solving
J. Maurice Rojas
TL;DR
The paper develops a polyhedral framework that links the Newton polytopes of multivariate polynomial systems to the number of complex roots. By combining lifted toric deformations, toric varieties, and mixed volumes, it provides self-contained proofs of Kushnirenko's unmixed theorem and Bernstein's mixed-volume bound, along with tight complexity results in the planar case. The introduction of mixed subdivisions and the Cayley trick gives a concrete, computable route to the mixed volume that underpins the general counting results. This approach yields both theoretical bounds on root counts and practical insights for structure-exploiting algorithms in algorithmic algebraic geometry. The work thus unifies polyhedral geometry with toric geometry to obtain robust, scalable results for solving polynomial systems.
Abstract
We give an elementary introduction to some recent polyhedral techniques for understanding and solving systems of multivariate polynomial equations. We provide numerous concrete examples and illustrations, and assume no background in algebraic geometry or convex geometry. Highlights include the following: (1) A completely self-contained proof of an extension of Bernstein's Theorem. Our extension relates volumes of polytopes with the number of connected components of the complex zero set of a polynomial system, and allows any number of polynomials and/or variables. (2) A near optimal complexity bound for computing mixed area -- a quantity intimately related to counting complex roots in the plane.