组合多项式单峰型性质的研究

Combinatorial polynomial is one of the basic objects in combinatorics. The unimodal property of combinatorial polynomials has been a hot topic in recent years. The project will focus on three important combinatorial polynomials, including:.1. W-polynomials on poset. We first generalize W-polynomials to multivariate polynomials according to the Eulerian polynomials. Then we study the unimodality and log-concavity of multivariate polynomials by means of the theory of complex analysises. Also, we will consider the q-SM property of W-polynomials, and explore the relation between the reality and the q-SM property by the theory of total positivity..2. Graph polynomials. We will investigate the unimodality of independence polynomials of trees by improving the methods including algebraic methods, combinatorial skills and linear transformations. We will show the reality of tau-polynomials, which are closely related to the chromatic polynomials, using the interlacing and compatible properties..3. Characteristic polynomials of the graph matrices. There have been many known results about the interlacing, the compatible and more generally the k-compatible properties of the eigenvalues of graph matrices. We will unify these results using the theory of symmetric matrices, and then simplify Marcus et al's proof of the interlacing families of characteristic polynomials.

组合多项式是组合数学中最基本的研究对象之一，组合多项式单峰型性质是近年来组合数学研究的热点。本项目拟研究三类组合多项式的单峰型性质：.1.偏序集中的W-多项式。比照Eulerian多项式将W-多项式推广到多元多项式，并借助复分析理论研究其单峰性和对数凹性。运用矩阵全正性理论研究W-多项式的q-Stieltjes moment(q-SM)性以及探索实零点性与q-SM性之间的关系。.2.图多项式。提升以往研究单峰性的代数方法、组合技巧和线性变换等方法，并结合递归关系研究Erdos等提出的树的独立多项式的单峰性猜想；借助零点交替性和相容性研究与色多项式相关的tau-多项式的实零点性。.3.图矩阵的特征多项式。在以往的研究中已得到各种图矩阵特征值满足交替性、相容性或更一般的k-相容性，但研究成果大都是孤立的，本项目将借助对称矩阵理论统一之前的结果，并给出Marcus等得到的交替族的简短证明。

组合多项式经常出现在数学的各个领域，有着很好的单峰型性质，本项目的主要研究成果包括：1.给出了Catalan-like数行发生函数序列Hankel行列式所满足的三项递归关系，由此统一得到了Catalan数、中心二项式系数和大小Schroder数行发生函数序列Hankel行列式所满足的递归关系。2.得到圈积上循环错排多项式和与Dowling格相关多项式的渐近正态性，以及循环错排多项式的单峰性和螺旋性。3.借助代数方法解决了超八面体群上两类对合多项式的gamma-正性的猜想，同时得到了无固定点对合多项式的单峰性和对称性。4.得到了一些零点分布性质的结果，例如，芘链的六隅体多项式的实零点性及渐近正态性，以及递归三角阵行发生函数序列反强Turan表达式的弱Hurwitz稳定性和三项递归多项式序列反强Turan表达式的实零点性，作为应用统一得到了Bell多项式、Dowling多项式、两类Chebyshev多项式、某些独立多项式和六隅体多项式的反强Turan表达式的零点分布性质。

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