Lorentzian多项式理论中的组合问题

围绕Petter Brändén与June Huh所建立的Lorentzian多项式理论，本项目拟在如下三个方面开展研究：.(1)运用经典的Littlewood-Richardson法则和陶哲轩等人发展的Hive模型等，给出June Huh等人得到的Koskta数的对数凹不等式的组合解释；.(2)结合组合数学方法与完全对数凹理论，推进June Huh等人提出的“正规化斜Schur多项式是Lorentzian多项式”这一猜想的研究；.(3)将稳定多项式中的容量等重要概念推广到Lorentzian多项式，从而应用Lorentzian多项式理论设计并优化计数拟阵基问题的渐近算法。.本项目的开展将为代数组合中的对数凹问题、计数算法问题的解决提供新角度，新方法，并将促进Lorentzian多项式理论与组合数学不同分支间的交叉融合，促进该前沿领域的发展。

Surrounding the theory of Lorentzian polynomials built by Petter Brändén and June Huh, this project mainly forces on the following three aspects:.(1)Using the classical Littlewood-Richardson rule and the Hive model developed by Terence Tao et al., we shall give a combinatorial interpretation of the log-concavity of Koskta number derived by June Huh et al..(2)We try to attack the conjecture of whether normalized skew Schur polynomials are Lorentzian polynomials by combinatorial methods and the theory of complete log-concave polynomials..(3)Some important concepts such as capacity in stable polynomials will be extended to Lorentzian polynomials. Then we shall employ the theory of Lorentzian polynomials on the design and optimization of asymptotic algorithms of counting bases of matroids..The execution of this project will provide new perspectives and methods for solving log-concavity problems and counting algorithm problems in algebraic combinatorics. We aim to establish the integration of the theory of Lorentzian polynomials and various branches of combinatorics and further promote the development of this frontier field.