Why Polyhedra Matter in Non-Linear Equation Solving

Details Why Polyhedra Matter in Non-Linear Equation Solving Why Polyhedra Matter in Non-Linear Equation Solving

2002-12-22

arxiv.org/abs/math/0212309v1

Mathematics

Algebraic Geometry

30 pages, 15 figures (26 ps or eps files), some in color. Paper corresponds to an invited tutorial talk delivered at a confere

Scientific paper

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.

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