多项式的实根性问题是组合数学中单峰型问题的一项重要研究内容，受到了很多组合数学家的关注。证明多项式实根性的方法涉及组合、代数、分析、几何等多个数学分支。其中，多项式的稳定性理论和零点交错理论近期取得了很多重要进展，被多次应用到纯粹数学和应用数学的很多领域，并在解决一些公开问题中发挥了巨大作用。本项目以解决组合多项式实根方面的几个公开猜想为研究目标，以单纯复形上的局部h-多项式、有限Coxeter群上的欧拉多项式和拟阵上的Kazhdan-Lusztig多项式为研究对象，研究这几类多项式的组合性质，并结合这些新的工具证明这几类多项式的实根性。

Many polynomials in combinatorics are known or conjectured to have only real zeros and have attracted interests of many combinatorists. Several methods from algebra, analysis, combinatorics and geometry are now available. Among them, the theory of stable polynomials and the theory of interlacing polynomials have been developed rapidly and have been applied to various areas of pure and applied fields of mathematics. Especially, these new tools have turned out to be an important ingredient for various long-standing open problems. In this project, we shall apply them to studying some open conjectures on the real-rootedness of certain combinatorial polynomials, including local h-polynomials of subdivisions of simplicial complexes, multivariate Eulerian polynomials for finite Coxeter groups and Kazhdan-Lusztig polynomials for matroids.