Symbolic Computation and Differential and Difference Equations

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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.