Symbolic Computation and Differential and Difference Equations

这个项目的目标是开发算法来确定微分和差分方程解的代数结构。 特别是，该项目旨在找到一个完整的有效算法来计算微分方程的伽罗瓦群和特定的算法来计算这些群中反映的方程的属性（例如， 有限项的可解性和低阶方程的可解性）。 精细的标准将寻求，这将允许一个特定的伽罗瓦组构造微分方程，并延长现有的解决方案连接线性代数群到任意线性代数群。 最近发展的伽罗瓦差分方程理论将适用于这些方程的类似问题。 特别是，有效的算法将被开发，以确定差分方程是否可以在有限的条件下求解，极大地推广了Petkovsek，Wilf和Zeilberger的工作。 此外，算法将被开发，以确定这样的方程的伽罗瓦群，并将获得这些方程的反问题的建设性解决方案。

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.