Convex bodies and algebraic equations on affine varieties
Convex bodies and algebraic equations on affine varieties
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发表时间：
2008-04
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通讯作者：
Kiumars Kaveh;A. Khovanskii
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作者：
Kiumars Kaveh;A. Khovanskii
给定仿射变量X和X上正则函数L的有限维向量空间，我们将一个凸体与（X， L）联系起来，使得它的体积负责L的一般函数系统的解的数量。这是牛顿多面体通常理论（涉及环面变量）的深远推广。作为应用，我们给出了代数几何（如Hodge指数定理）和凸几何（如Alexandrov-Fenchel不等式）中一些著名定理的新的、简单的、透明的证明。我们的主要工具是经典的希尔伯特理论在投影空间的次变异度（代数几何）和布伦-闵可夫斯基不等式（凸几何）。
Given an affine variety X and a finite dimensional vector space of regular functions L on X, we associate a convex body to (X, L) such that its volume is responsible for the number of solutions of a generic system of functions from L. This is a far reaching generalization of usual theory of Newton polytopes (which is concerned with toric varieties). As applications we give new, simple and transparent proofs of some well-known theorems in both algebraic geometry (e.g. Hodge Index Theorem) and convex geometry (e.g. Alexandrov-Fenchel inequality). Our main tools are classical Hilbert theory on degree of subvarieties of a projective space (in algebraic geometry) and Brunn-Minkowski inequality (in convex geometric).