论文标题
在ehrhart的多项式上
On the Ehrhart Polynomial of Minimal Matroids
论文作者
论文摘要
我们为Ehrhart的多项式提供了一个公式，该ehrhart多项式的尺寸$ n $和等级$ k $的基础数量最少，也称为最小矩阵。我们证明他们的多面是ehrhart的正面和$ h^*$ - 实用的（因此是单峰）。我们证明，电路光平面松弛的运行与最小的矩形和矩阵多型分区有关，并且还保留了ehrhart的阳性。我们陈述了两个猜想：实际上，所有矩形都是$ h^*$ - 实用的，并且Ehrhart多项式的系数和固定等级和心脏的连接基曲线的界限都受到相应的最小矩形和相应统一的均匀的Matroid的界限。
We provide a formula for the Ehrhart polynomial of the connected matroid of size $n$ and rank $k$ with the least number of bases, also known as a minimal matroid. We prove that their polytopes are Ehrhart positive and $h^*$-real-rooted (and hence unimodal). We prove that the operation of circuit-hyperplane relaxation relates minimal matroids and matroid polytopes subdivisions, and also preserves Ehrhart positivity. We state two conjectures: that indeed all matroids are $h^*$-real-rooted, and that the coefficients of the Ehrhart polynomial of a connected matroid of fixed rank and cardinality are bounded by those of the corresponding minimal matroid and the corresponding uniform matroid.
