Computing galois groups for certain classes of ordinary differential equations

by 1969- Berman, Peter Hillel

BERMAN, PETER HILLEL. Computing Galois Groups for Certain Classes of Ordinary Differential Equations. (Under the direction of Michael Singer.) 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 C = ¯ Q, 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.