Piecewise polynomial functions, convex polytopes and enumerative geometry
Piecewise polynomial functions, convex polytopes and enumerative geometry
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分段多项式函数、凸多面体和枚举几何
DOI：
10.4064/-36-1-25-44
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发表时间：
1996
期刊：
影响因子：
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通讯作者：
M. Brion
中科院分区：
文献类型：
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作者：
M. Brion
0. 介绍。本文探讨了其标题的一些对象之间的联系。它是基于McMullen的多面体代数的一种新方法，以及它与环变异的等变上同调的关系。特别地，我们给出了Fulton和Sturmfels最近的一个结果的另一个证明，该结果识别了具有光滑、完全环面嵌入的所有Chow环的直接极限的多面体代数（见[14]）。另一方面，我们得到了经典Bezout定理的一个版本，它适用于任何球面齐次空间。这推广了Bernstein和Kouchnirenko的一个定理：d维环面一般位置上的d超曲面的公点数是d！乘以相关牛顿多面体的混合体积（参见[2]，[18]和[13]5.5）。给定一个有序域K上的有限维向量空间V，对偶空间V *中的凸多面体与V上的分段线性凸函数之间存在着众所周知的对应关系。也就是说，对于任何凸多面体，我们将其支持函数关联起来。用R表示由V上的连续函数代数中所有凸多边形的支持函数所生成的代数。本文第一部分研究了K为有理数域时的代数R。结果是（见1.5）R是V上的连续分段多项式函数的代数；特别地，R包含多项式函数的代数。我们证明了V上任意坐标函数的选择在R中定义了一个正则序列；此外，由V *生成的理想R的商与McMullen的有理多面体代数同构（见1.3,1.5；我们的证明是基于Morelli的工作，见[21]）。更一般地说，由V *生成的理想的幂R的商，与最近由McMullen， Pukhlikov和Khovanskii考虑的多体代数的高级版本同构；参见[20]，[24]。事实上，我们研究代数R作为它的子代数RΣ的直接极限，这些子代数是由关于一个固定扇形Σ的分段多项式函数组成的。对这样的粉丝来说
0. Introduction. This paper explores some of the connections between the objects of its title. It is based on a new approach to McMullen’s polytope algebra, and on its relation with equivariant cohomology of toric varieties. In particular, we give another proof of a recent result of Fulton and Sturmfels, which identifies the polytope algebra with the direct limit of all Chow rings of smooth, complete torus embeddings (see [14]). On the other hand, we obtain a version of the classical theorem of Bezout, which holds in any spherical homogeneous space. This generalizes a theorem of Bernstein and Kouchnirenko: The number of common points to d hypersurfaces in general position in a d-dimensional torus is d! times the mixed volume of the associated Newton polytopes (see [2], [18] and also [13] 5.5). Given a finite-dimensional vector space V over an ordered field K, there is a wellknown correspondence between convex polytopes in the dual space V ∗ and piecewise linear convex functions on V . Namely, to any convex polytope, we associate its support function. Denote by R the algebra generated by the support functions of all convex polytopes, in the algebra of continuous functions on V . In the first section of this paper, we study the algebra R when K is the field of rational numbers. It turns out (see 1.5) that R is the algebra of continuous, piecewise polynomial functions on V ; in particular, R contains the algebra of polynomial functions. We prove that any choice of coordinate functions on V defines a regular sequence in R; moreover, the quotient of R by the ideal generated by V ∗ is isomorphic to the rational polytope algebra of McMullen (see 1.3, 1.5; our proof is based on work of Morelli, see [21]). This explains the non-trivial grading of the polytope algebra, by the obvious grading of R. More generally, the quotients of R by powers of the ideal generated by V ∗, are isomorphic to the higher versions of the polytope algebra, considered recently by McMullen, Pukhlikov and Khovanskii; see [20], [24]. In fact, we study the algebra R as the direct limit of its subalgebras RΣ consisting of functions which are piecewise polynomial with respect to a fixed fan Σ. To such a fan is