超平面配置的特征多项式推广了图的色多项式，也是超平面配置理论中最重要的一个不变量. 2012年,Huh利用超平面配置的理论证明了Read在1968年提出的色多项式系数单峰性猜想. . 我们此次研究计划将围绕Read当年提出的另一个问题，什么样的多项式是图的色多项式? 类似于Huh的处理方式,我们将在超平面配置的理论框架下,从多个角度对其特征多项式进行研究,然后将研究结果应用到图的色多项式来处理Read所提的问题。. 为此，此次计划的研究内容包含以下四个方面，. 1. 超平面配置的交半格的同构分类及其对应的特征多项式; . 2. 超平面配置特征多项式系数的组合关系式及相应的组合或几何解释;. 3. 将线性超平面配置的算术性质推广至一般的超平面配置及Erhart拟多项式上;. 4. Tutte多项式的组合计数解释.

Characteristic polynomials of hyperplane arrangements generalize chromatic polynomials of graphs and become the most important invariant of hyperplane arrangements. As an application of hyperplane arrangements,in 2012,Huh proved the unimodality conjecture on the coefficients of chromatic polynomials which was proposed by Read in 1968. . In this research proposal,our motivation is the other question raised by Read in 1968,which polynomial is a chromatic polynomial of some graph? Similar to Huh's way,we will work on the hyperplane arrangement to study its characteristic polynomial through several different ways, and then apply the results to the chromatic polynomial in Read's question. . To this end,the poposal will cover the following four research topics,. 1. classifying isomorphic intersection semi-lattices of hyperplane arrangements and determining the corresponding characteristic polynomials;. 2. finding combinatorial formula for coefficients of characteristic polynomials and exploring their combinatorial or geometric interpretations;. 3. extending the arithmetic properties from central hyperplane arrangements to affine cases and Erhart quasi-polynomials;. 4. providing the combinatorial interpretation of Tutte polynomials.