论文标题
共振安排及其贝蒂数字的普遍性
The Universality of the Resonance Arrangement and its Betti Numbers
论文作者
论文摘要
共振安排$ \ MATHCAL {a} _n $是超平面的安排，它在$ \ Mathbb {r}^n $中都以正常向量为$ \ Mathbb {r}^n $。它是辫子布置的伴随，也称为全烟安排。本文的第一个结果表明，任何理性的超平面布置都是一些足够大的共振安排的次要。 它的腔室是代数几何形状中多项式的区域，作为数学物理学中的概括性功能，以及在经济学中应用的最大不平衡家庭。计算任何真实布置的室数的一种方法是通过其特征多项式的系数，称为betti数字。我们表明，共振布置的贝蒂数是由第二类的stirling数的固定组合确定的。最后，我们为共振安排的前两个非平凡的Betti数字开发了确切的公式。
The resonance arrangement $\mathcal{A}_n$ is the arrangement of hyperplanes which has all non-zero $0/1$-vectors in $\mathbb{R}^n$ as normal vectors. It is the adjoint of the Braid arrangement and is also called the all-subsets arrangement. The first result of this article shows that any rational hyperplane arrangement is the minor of some large enough resonance arrangement. Its chambers appear as regions of polynomiality in algebraic geometry, as generalized retarded functions in mathematical physics and as maximal unbalanced families that have applications in economics. One way to compute the number of chambers of any real arrangement is through the coefficients of its characteristic polynomial which are called Betti numbers. We show that the Betti numbers of the resonance arrangement are determined by a fixed combination of Stirling numbers of the second kind. Lastly, we develop exact formulas for the first two non-trivial Betti numbers of the resonance arrangement.
