Symbolic Computation and Differential and Difference Equations

该项目的目标是开发确定微分方程解和差分方程解的代数结构的算法。特别是，该项目试图找到一个完整有效的算法来计算微分方程式的伽罗瓦群，并寻找具体的算法来计算这些群中所反映的方程的性质(例如，有限项的可解性和低阶方程的可解性)。将寻求精确的准则，以允许构造具有指定伽罗瓦群的微分方程，并将连通线性代数群的现有解扩展到任意线性代数群。最近发展起来的伽罗瓦差分方程组理论也将应用于这些方程的类似问题。特别是，将开发有效的算法来确定差分方程组是否可以有限项求解，这将极大地推广Petkovsek，Wilf和Zeilberger的工作。此外，还将开发确定此类方程的伽罗瓦群的算法，并将获得这些方程的反问题的构造性解。

The goal of this project is to develop algorithms to determine the algebraic structure of solutions of differential and difference equations. In particular, the project seeks to find a complete efficient algorithm to compute the Galois groups of differential equation and specific algorithms to compute properties of the equations as reflected in these groups (e.g., solvability in finite terms and solvability in terms of lower order equations). Refined criteria will be sought that will allow one to construct differential equations with a specified Galois group and extend the existing solution for connected linear algebraic groups to arbitrary linear algebraic groups. The recently developed Galois theory of difference equations will be applied to similar problems for these equations as well. In particular, effective algorithms will be developed to determine if difference equations can be solved in finite terms, greatly generalizing the work of Petkovsek, Wilf, and Zeilberger. Furthermore, algorithms will be developed to determine the Galois groups of such an equation, and a constructive solution of the inverse problem for these equations will be obtained.