Computing Galois Groups for Certain Classes of Ordinary Differential Equations

Abstract

As of now, it is an open problem to find an algorithmthat computes the Galois group G of an arbitrary linear ordinary differential operator L in C(x)[D]. We assume thatC is a computable, characteristic-zero,algebraically closed constant field with factorization algorithm.In this dissertation, we present new methods forcomputing differential Galois groups in two special cases.An article by Compoint and Singer presents a decision procedure to compute G in case L is completely reducible or, equivalently, G is reductive. Here, we present the results of an article by Berman and Singerthat reduces the case of a productof two completely reducible operators to thatof a single completely reducible operator;moreover, we give an optimization of that article's core decision procedure.These results rely on results from cohomologydue to Daniel Bertrand.We also give a set of criteria to compute the Galois group of a differential equation of the formy⁽³⁾ + ay' + by = 0, a, b in C[x].Furthermore, we present an algorithm to carry out this computation in case C is the field of algebraic numbers.This algorithm applies the approach used inan article by M. van der Put to study order-two equations with one or two singularpoints. Each step of the algorithm employs a simple, implementable test based on some combination of factorization properties, properties of associated operators,and testing of associated equations for rational solutions. Examples of the algorithm and a Maple implementation writtenby the author are provided.