基于符号计算的代数微分和差分方程的理论及算法研究

代数微分和差分方程组是数学，计算机及其相关领域的重要研究课题。符号计算是研究代数微分和差分方程理论和算法的强有力方法。我们研究与此相关的两个基本问题：1.线性微分和差分方程的奇点消去问题； 2. 非线性代数微分和差分方程的闭形式解问题。对于第一个问题，Ore算子给出了线性微分和差分方程统一的代数抽象。我们将利用Groebner基和Ore代数理论估计Ore算子奇点消尽算子阶的上界，从而给出Ore算子奇点消去的完整算法。对于第二个问题，我们将通过代数曲线的参数化和消去理论估计出一阶代数差分方程强一般有理解次数的上界，以此给出计算其强一般有理解的完整算法。此外，我们将给出微分多项式向量导数的展开公式从而设计计算一类代数微分方程组的幂级数解的算法。作为应用，我们将把相应的研究成果应用于组合数学，特殊函数，代数统计，密码编码，以及含参线性微分和差分方程Galois理论中的重要计算问题中。

Algebraic differential and difference equations are important research topics in mathematics, computer science, and related areas. Symbolic computation is a powerful approach for studying theory and algorithms of algebraic differential and difference equations. We study two fundamental problems on this topic: 1. Desingularization of linear differential and difference equations; 2. Closed form solutions of nonlinear algebraic differential and difference equations. For the first problem, Ore operators form a common algebraic abstraction of linear differential and difference equations. We will utilize Groenber bases and Ore algebra theories to give an order bound of desingularized operators of Ore operators, and thus give a complete algorithm for desingularization of Ore operators. For the second problem, we will first use parametrization of algebraic curves and elimination theory to derive a degree bound of strong general solutions of first-order algebraic difference equations, and therefore present a complete algorithm for computing its strong general solutions. Moreover, we will give an expansion formula for derivatives of differential polynomial vectors so that we can design an algorithm for computing formal power series solutions of a class of systems of algebraic ordinary differential equations. As applications, we will apply the corresponding results to important computational problems in combinatorics, special functions, algebraic statistics, coding and cryptography, and the Galois theory of parameterized linear differential and difference equations.