The Study Of Multivariate Polynomial Sequences With Real Stability

Zeros distribution problem of polynomials is not only an important topic in combinatorics,but also one of the most primary problems in mathematics.The zeros distribution problems of polynomials are closely related to many classical mathematical problems,such as the four-color problem.Stability and real-rootedness are two important topics in the zeros distribution problems of polynomials.With the development of combinatorics,the stability and real stability of multivariate polynomials are hot topics in combinatorics.We can solve the problem of univariate polynomials by the theory of multivariate polynomials.In this dissertation,we mainly prove the real stability or the proper position property of some multivariate polynomials in a unified approach.And we also prove the stability of the multivariate polynomials by the context-free grammar and operator of stability preserving.The details of this dissertation are as follows:In the first part,we mainly discuss the real stability or the proper position property of multivariate polynomials in a unified approach.On the one hand,we give sufficient conditions to prove the real stability of the multivariate polynomial sequences.And we get some results for the stability of multivariate polynomials.As applications,we get the real stability of multivariate Eulerian polynomials,multivariate Bell polynomials,multivariate polynomials over Stirling permutations.On the other hand,we also give sufficient conditions to prove the proper position property of multivariate polynomial sequences.As applications,we get the proper position property of multivariate orthogonal polynomials and multivariate matching polynomials.In addition,we also obtain a multivariate generalization of Fisk’s result by presenting a 2 × 2 matrix,which preserves the proper position property.The second part mainly gives the combinatorial interpretation and recursive relations of multivariate polynomials by context-free grammars,and proves the stability of multivariate polynomials by finding the operator of stability preserving.Firstly,we prove the real stability of multivariate polynomials over the Jacobi-Stirling permutations and the multivariate Lah polynomials in a unified approach.Secondly,we give the stability of these multivariate polynomials by context-free grammars and the operator of stability preserving.Finally,we also give the stability of Eulerian polynomials of type and multivariate polynomials over the Stirling permutations in a similar way.