递归多项式的根几何及其应用

Root geometry reveals deep information of polynomials. Many theoretical results and applications on root geometry have been published on top journals such as Ann. of Math. Nowadays, root geometry is one of the most popular topics in mathematical research. Real-rootedness is an important part of root geometry, with log-concavity as an implication, it occurs naturally in many mathematical disciplines such like probability and statistics, algebra, analysis and geometry...In this program, we focus on polynomial sequences satisfying a recurrence relation. Our goal is to figure out the fundamental structure of root geometry of such polynomial sequences. For those with only real roots, we study the ``no-root-intervals'', ``convergence intervals'', and ``interlacing intervals'', and use the results to solve the problem of iterated 3-wheels in the LCGD conjecture. We also consider the relation between characteristic equations of the recurrence and the root geometry. In particular, we develop the theory for the root geometry of recursive polynomials of order two. Meanwhile, we study the fundamental structure of root geometry of recursive polynomials which are not real-rooted. In this way we explore the relation between root geometry and log-concavity of the same polynomial sequence...The theoretical results obtained in this program will enrich the theory of root geometry, develop the research of genus distribution in the inter-discipline of topological graph theory and enumerative combinatorics. Moreover, the study will offer more theoretical support and application examples to the application of root geometry in other mathematical branches.

根几何揭示多项式深层次的信息，其许多相关成果与应用发表在Ann. of Math.等顶级数学期刊，是当今数学界的前沿研究课题之一。实根性是根几何的一个重要组成部分，它与它蕴含的对数凹性自然地出现于概率与统计、代数、分析、几何等许多数学分支。..本项目着眼于满足递归关系的多项式序列，旨在研究其根几何的基本结构。对于实根递归多项式序列，项目申请人利用组合数学的分析方法与技巧，研究确定该序列的“无根区间”、“收敛区间”和“交错区间”，将之应用于解决LCGD猜想中的迭代三轮图问题等。本项目将明确递归关系的特征方程与根几何的内在关系，特别是完善二阶递归多项式的根几何理论。同时，项目研究非实根递归多项式的根几何结构，探索它与对数凹性的联系。..本项目的研究成果将丰富多项式的根几何理论，推进拓扑图论与计数组合交叉领域中亏格多项式等对象的研究，并为根几何在其它数学分支中的应用提供更多的理论支持和应用实例。

根几何揭示多项式深层次的信息，是数学界的前沿研究课题之一。实零点性是根几何的一个重要组成部分，它与它蕴含的对数凹性自然地出现于概率与统计、代数、分析、几何等许多数学分支。项目研究满足递归关系的多项式序列的根几何，对实零点序列确定其无根区间、收敛区间和交错区间。项目的重要结果包括明确了递归关系的特征方程与根几何的内在关系、利用组合不等式解决了某些二阶递归多项式的根几何结构问题，并给出图的色对称函数的Schur系数的组合解释。