标题： 加权格点的Ehrhart函数 显示英文标题

标题： Ehrhart Functions of Weighted Lattice Points

摘要： 本文研究了三种不同的为凸多面体的格点分配权重的方法，并讨论了由此产生的加权Ehrhart函数及其生成函数和相关环的代数和组合性质。 这些将分别称为$q$-加权、$r$-加权和$s$-加权Ehrhart函数。 我们研究的关键问题是\emph{加权Ehrhart级数何时为有理函数，哪些经典的Ehrhart理论性质会被保留？ 以及，抽象形式幂级数何时为某个多面体的Ehrhart环的Hilbert级数？}我们证明了关于加权Ehrhart系数量的推广，以及显示$h^*$-系数的$q$-加权Ehrhart级数，并展示$q$-和$s$-加权Ehrhart互反定理。然后，我们展示了$q$-和$r$-加权Ehrhart环是权提升多面体的（经典）Ehrhart环。 显示英文摘要

摘要： This paper studies three different ways to assign weights to the lattice points of a convex polytope and discusses the algebraic and combinatorial properties of the resulting weighted Ehrhart functions and their generating functions and associated rings. These will be called $q$-weighted, $r$-weighted, and $s$-weighted Ehrhart functions, respectively. The key questions we investigate are \emph{When are the weighted Ehrhart series rational functions and which classical Ehrhart theory properties are preserved? And, when are the abstract formal power series the Hilbert series of Ehrhart rings of some polytope?} We prove generalizations about weighted Ehrhart $h^*$-coefficients of $q$-weighted Ehrhart series, and show $q$- and $s$-weighted Ehrhart reciprocity theorems. Then, we show the $q$- and $r$-weighted Ehrhart rings are the (classical) Ehrhart rings of weight lifting polytopes.