Why Polyhedra Matter in Non-Linear Equation Solving
Why Polyhedra Matter in Non-Linear Equation Solving
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为什么多面体在非线性方程求解中很重要
DOI：
10.1090/conm/334/05987
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发表时间：
2002
期刊：
影响因子：
--
通讯作者：
M. Rojas
中科院分区：
文献类型：
--
作者：
M. Rojas
本文对近年来用于理解和求解多元多项式方程组的多面体技术作了初步介绍。我们提供了许多具体的例子和插图，并假设在代数几何或凸几何背景。(1) Bernstein的Theo- rem扩展的一个完全自包含证明。我们的扩展将多面体的体积与多项式系统的复零集的连通分量的数量联系起来，并且允许任意数量的多项式和/或变量。(2)计算混合面积的近最优复杂度界——一个与计算平面上的复根密切相关的量
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 Theo- rem. 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 quan- tity intimately related to counting complex roots in the plane