我们主要利用符号计算的方法研究组合多项式的单峰型问题。组合多项式的单峰型问题是近年来组合数学研究的前沿领域，而符号计算方法将为其研究提供新的方法与工具。本项目将重点研究以下问题：.1.组合多项式的（高阶）q-对数凹凸性研究。将通过q-级数理论、生成函数及成熟的程序包，实现对多项式的高阶对数凹凸性质的机械化证明，解决几个公开问题；.2.特殊函数的渐进性质。将利用渐进估计、Taylor展式、柱形代数分解等方法，研究特殊函数序列的渐近单峰型和单峰型性质，并解决几个相关猜想；.3.多变元多项式的单峰性研究。将借助对称函数理论等实现对该类问题的化简并实现自动化证明，在研究中我们将以Mathematica等数学软件为工具，综合运用经典方法和计算机代数相结合展开交叉研究。我们力争给出若干创新性的解题方法，解决一些猜想，并在相关研究上取得重要进展。

We focus on proving unimodality of combinatorial polynomials by using the method of symbolic computation. The unimodality of combinatorial polynomials have been the forefront of combinatorics..The symbolic computation will provide new ideas and tools for the research. In this project, we will mainly consider the following problems:.1. The (higher order) q-log-convexity and q-log-concavity of combinatorial polynomials. We will utility the q-series theory，generating functions and mature program packages, accomplish the automatic proof of higher-order log-behavior of polynomials, and then several open problems will be solved;.2. Asymptotic properties of special functions. Based on the symbolic computation of asymptotic estimations, Taylor expansion, and cylindrical algebraic decomposition, we will study the asymptotic unimodality type and unimodality of special functional sequences, and solve several related conjectures;.3. Unimodality of multivariate combinatorical polynomials. We will utility the theory of symmetric function to realise the symplication and automatic proof of this type of problems. In the research, we will use mathematical software such as Mathematica as a tool, and combine classical methods and computer algebra to do cross-research. We strive to provide some creative methods, solve related conjectures and obtain some significint results in related fields.