Logarithmic Concavity Of Kazhdan-Lusztig Polynomials On Some Matroid

The log-concavity of combinatorial sequences is one of the hot topics in combinatorial mathematics research in recent years,and has very important applications and significance in combinatorial mathematics.Elias,Proudfoot,and Wakefield associated to every matroid M a polynomial with integer coefficients.The polynomial are usually called the KazhdanLusztig polynomial of M.They proposed a conjecture which states that the coefficients of these polynomials are always non-negative.This conjecture was recently confirmed by Braden,Huh,Matherne,Proudfoot,and Wang.Elias,Proudfoot,and Wakefield also conjectured the log-concavity of these polynomials.This conjecture is still open.In this paper,we mainly study the Kazhdan-Lusztig polynomials of thagomizer matroids,Z-polynomials and γ-polynomials of uniform matroids,and sparse paving matroids.Then we use computer algebra approach such as Zeilberger algorithm and HolonomicFunctions to prove the log-concavity and ultra log-concavity of combinatorial sequences.This thesis is organized as follows.The first chapter is the introduction,which first introduces the research background and history of related properties such as unimodal and log-concavity,as well as the symbols and basic concepts related to the paper,such as unimodal,log-concavity,and ultra log-concavity.Secondly,we briefly introduce the Kazhdan-Lusztig theory of matroids.In the second chapter,we introduce the background of computer algebra approach such as Gosper algorithm and Zeilberger algorithm,HolonomicFunctions package,Mgfun package,and give some examples of computer algebra approach of recursive relations.At the same time,we also introduce the use of CylindricalDecomposition.In the third chapter,we study the log-concavity of Kazhdan-Lusztig polynomials of thagomizer graph matroids.We mainly use the Mathematica and Zeilberger algorithm to obtain the recurrence relation.Secondly,by estimating upper and lower bounds,we prove the log-concavity of the coefficients of the Kazhdan-Lusztig polynomials of the thagomizer graph matroids.In the fourth chapter,we study the ultra log-concavity of Z-polynomial and γ-polynomial of uniform matroids,γ-positivity of uniform matroids and Sparse paving matroids.It mainly uses Mathematica and HolonomicFunctions package to get recurrence relation,CylindricalDecomposition command to deal with the inequality.