# Computing galois groups for certain classes of ordinary differential equations

Abstract (Summary)

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.
Bibliographical Information:

Advisor:

School:North Carolina State University

School Location:USA - North Carolina

Source Type:Master's Thesis

Keywords:north carolina state university

ISBN:

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