为什么多面体在非线性方程求解中至关重要
Why Polyhedra Matter in Non-Linear Equation Solving
AI总结：
本文以初等方式介绍多面体技术在多元多项式方程组求解中的应用，给出Bernstein定理的推广证明及混合面积计算的近最优复杂度界。
摘要：
我们给出了一个关于近期用于理解和求解多元多项式方程组的多面体技术的初等介绍。我们提供了大量具体例子和插图，并假设读者没有代数几何或凸几何背景。亮点包括：（1）一个完全自包含的Bernstein定理推广证明。我们的推广将多胞体的体积与多项式系统复零点集的连通分量数联系起来，并允许任意数量的多项式和/或变量。（2）一个计算混合面积——一个与平面中复根计数密切相关的量——的近乎最优复杂度界。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
