通过判别式排列（Discriminantal arrangement）探讨广义西尔维斯特（Sylvester）问题与果园（Orchard）问题
The Generalized Sylvester's And Orchard Problems Via Discriminantal arrangement
AI总结：
本文揭示了判别式排列$\mathcal{B}(n, k,\mathcal{A})$组合学变化与广义Sylvester问题和orchard问题中点构型分类的联系，为解决这些未解问题提供了新视角。
摘要：
1989年，Manin和Schechtman定义了与$k$维空间中$n$个超平面的通用排列$\mathcal{A}$相关联的判别式排列$\mathcal{B}(n, k,\mathcal{A})$。Crapo早在1985年就以电路几何（geometry of circuits）的名称引入了等价概念。虽然这两篇论文主要关注当$\mathcal{A}$变化时$\mathcal{B}(n, k,\mathcal{A})$具有恒定组合学的情形，但事实证明，$\mathcal{B}(n, k,\mathcal{A})$组合学发生变化的情形同样十分有趣，因为它能对$k$维空间中点的特殊构型进行分类。本文提供了一个实例来阐明这一事实，揭示了著名的广义Sylvester问题和orchard问题与$\mathcal{B}(n, k,\mathcal{A})$组合学之间的联系。特别地，我们指出了这种联系如何有助于解决这些古老但仍未解决的问题。
英文摘要：
In 1989 Manin and Schechtman defined the discriminantal arrangement $\mathcal{B}(n, k,\mathcal{A})$ associated to a generic arrangement $\mathcal{A}$ of $n$ hyperplanes in a $k$-dimensional space. An equivalent notion was already introduced by Crapo in 1985 with the name of geometry of circuits. While both those papers were mainly focused on the case in which $\mathcal{B}(n, k,\mathcal{A})$ has a constant combinatorics when $\mathcal{A}$ changes, it turns out that the case in which the combinatorics of $\mathcal{B}(n, k,\mathcal{A})$ changes is quite interesting as it classifies special configurations of points in the $k$-dimensional space. In this paper we provide an example of this fact elucidating the connection between the well known generalized Sylvester's and orchard problems and the combinatorics of $\mathcal{B}(n, k,\mathcal{A})$. In particular we point out how this connection could be helpful to address those old but still open problems.
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