Computing Galois Groups for Certain Classes of Ordinary Differential Equations

by Berman, Peter Hillel

Abstract (Summary)

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''' + 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.

Bibliographical Information:

Advisor:Michael Singer; Ronald Fulp; Kailash Misra; Larry Norris

School:North Carolina State University

School Location:USA - North Carolina

Source Type:Master's Thesis



Date of Publication:07/25/2001

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