Ehrhart Theory for Lawrence Polytopes and Orbifold Cohomology of Hypertoric Varieties

Faithful tropicalization of hypertoric varieties

The hypertoric variety $\mathfrak{M}_{\mathcal{A}}$ defined by an affine arrangement $\mathcal{A}$ admits a natural tropicalization, induced by its embedding in a Lawrence toric variety. We explicitly describe the polyhedral structure of this tropicalization. Using a recent result of Gubler, Rabinoff, and Werner, we prove that there is a continuous section of the tropicalization map.

源于 Ehrhart 理论的多胞体代数体积

Algebraic Volume for Polytope Arise from Ehrhart Theory

本文基于 Ehrhart 理论将多胞体提升为锥，利用代数组合学技术提出一种仅需符号单纯锥分解的体积算法，并给出全新的原始-对偶方法，解决了现有方法对简单或单纯多胞体不切实际的问题。

Volumes and Tangent Cones of Matroid Polytopes

De Loera et al. 2009, showed that when the rank is fixed the Ehrhart polynomial of a matroid polytope can be computed in polynomial time when the number of elements varies. A key to proving this is the fact that the number of simplicial cones in any triangulation of a tangent cone is bounded polynomially in the number of elements when the rank is fixed. The authors speculated whether or not the Ehrhart polynomial could be computed in polynomial time in terms of the number of bases, where the number of elements and rank are allowed to vary. We show here that for the uniform matroid of rank $r$ …

论Arthur迹公式的组合学、凸多面体与环面簇

On Combinatorics of the Arthur Trace Formula, Convex Polytopes, and Toric Varieties

本文从扇与多面体出发构造组合截断函数，证明其积分的收敛性与多项式性，推广了Arthur迹公式、Ehrhart多项式及Langlands引理，并给出环面簇上的几何解释。