Computing Galois groups for certain classes of ordinary differential equations.

摘要

As of now, it is an open problem to find an algorithm that computes the Galois group G of an arbitrary linear ordinary differential operator L ∈ $C$(x)[D]. We assume that $C$ is a computable, characteristic-zero, algebraically closed constant field with factorization algorithm. In this dissertation, we present new methods for computing 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 Singer that reduces the case of a product of two completely reducible operators to that of a single completely reducible operator; moreover, we give an optimization of that article's core decision procedure. These results rely on results from cohomology due to Daniel Bertrand.; We also give a set of criteria to compute the Galois group of a differential equation of the form y(3) + ay ′ + by = 0, a, b ∈ $C$[x]. Furthermore, we present an algorithm to carry out this computation in case , the field of algebraic numbers. This algorithm applies the approach used in an article by M. van der Put to study order-two equations with one or two singular points. 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 written by the author are provided.