本项目主要研究复域上的二阶非线性差分方程的分类问题和指数多项式零点分布。首先，差分方程是微分方程的离散化，为了模拟二阶非线性微分方程的分类，即Painleve方程，本项目预计对二阶非线性差分方程进行分类，寻找到对应的差分Painleve方程，这将推动复域上差分方程的进展。其次，注意到线性差分方程的解对应了指数多项式的零点，我们也将研究指数多项式的零点分布，特别是一阶指数多项式零点的渐进分布，这将推动Shapiro猜想的进展。

In this project, we study the classification of second order nonlinear difference equations and zero distributions of exponential polynomials. First, differential equations a continuous form of difference equations. In order to analogy the classification of second order nonlinear differential equations Painleve equation. We expect to classify two order nonlinear difference equations, seeking for difference Painleve equations, which will promote the progress of difference equations in complex plane. Secondly, the solutions of linear difference equations.are related to the zeros of exponential polynomials. In this project will find the zero distributions of the exponential polynomials, especially the asymptotic zero distributions of the first order exponential polynomials, the results on zeros distributions of first order exponential polynomial will promote the progress of the Shapiro conjecture.