Symbolic Computation and Differential and Difference Equations

Michael F.辛格建议研究开发有效的算法来确定微分方程和差分方程解的代数结构。特别是调查人员提出，以找到有效的算法来计算伽罗瓦群的大型类的微分方程和努力寻找一个通用的算法来计算伽罗瓦群的任何线性微分方程。他还提出要找到有效的算法来计算这些组中反映的方程的属性（例如，有限项可解性和低阶方程可解性），并将这些算法应用于Hamilton系统的可积性问题。研究者还将使用这些算法给出有效的方法来确定代数方程的性质（例如，绝对不可约，伽罗瓦群的计算）。他建议找到完善的标准，将允许一个构造微分方程与一个指定的伽罗瓦组和延长他的解决方案的反问题的连接线性代数群的任意线性代数群。他将适用于他最近开发的伽罗瓦理论的差分方程类似的问题，这些方程以及。特别是他建议完善的算法，以确定是否差分方程可以解决有限的条款，并延长这一q-差分方程，大大推广的工作Petkovsek，Wilf和Zeilberger，制定算法，以确定伽罗瓦组这样一个方程，并给予建设性的解决方案的反问题，这些方程。

Michael F. Singer proposes research to develop efficient algorithms to determine the algebraic structure of solutions of differential and difference equations. In particular the investigator proposes to find efficient algorithms to compute the Galois groups for large classes of differential equations and work towards finding a general algorithm to calculate the Galois group of any linear differential equation. He proposes to also find efficient 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) and apply these algorithms to integrability problems of Hamiltonian systems. The investigator will also use these algorithms to give efficient methods to determine properties of algebraic equations (e.g., absolute irreducibility, calculation of Galois groups). He proposes to find refined criteria that will allow one to construct differential equations with a specified Galois group and extend his solution of the inverse problem for connected linear algebraic groups to arbitrary linear algebraic groups. He will apply his recently developed Galois theory of difference equations to similar problems for these equations as well. In particular he proposes to refine the algorithms to determine if difference equations can solved in finite terms and extend this to q-difference equations, greatly generalizing the work of Petkovsek, Wilf and Zeilberger, develop algorithms to determine the Galois group of such an equation and give a constructive solution of the inverse problem for these equations.