Symbolic Computation and Differential and Difference Equations

Michael F. Singer建议研究开发有效的算法来确定微分和差分方程解的代数结构。特别地，研究者建议寻找一种有效的算法来计算大型微分方程的伽罗瓦群，并努力寻找一种通用的算法来计算任何线性微分方程的伽罗瓦群。他还建议寻找有效的算法来计算这些组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），并将这些算法应用于哈密顿系统的可积性问题。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，绝对不可约性，伽罗瓦群的计算）。他提出了一种改进的准则，使人们能够构造具有特定伽罗瓦群的微分方程，并将他的连通线性代数群反问题的解推广到任意线性代数群。他将把他最近发展的差分方程伽罗瓦理论应用于这些方程的类似问题。他特别提出了改进差分方程是否可以有限项解的算法，并将其推广到q差分方程，极大地推广了Petkovsek、Wilf和Zeilberger的工作，发展了确定这类方程的伽罗瓦群的算法，并给出了这些方程的反问题的建设性解。

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.