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.
