Galois theory of linear ODEs

157 Pages·2001·1.03 MB·English

by  Berman P.H.

Abstract
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(cid:48) +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.
COMPUTING GALOIS GROUPS FOR CERTAIN CLASSES OF
ORDINARY DIFFERENTIAL EQUATIONS
by
PETER HILLEL BERMAN
A dissertation submitted to the Graduate Faculty of
North Carolina State University
in partial fulfillment of the
requirements for the Degree of
Doctor of Philosophy
MATHEMATICS
Raleigh
2001
APPROVED BY:
Chair of Advisory Committee
In memory of Dan Berman
ii
Biography
Peter Hillel Berman was born in Minneapolis, Minnesota, on August 21, 1969. He
graduated from Saint Louis Park Senior High School in Saint Louis Park, Minnesota, in
1987. He received an A.B. degree in mathematics from Washington University in Saint
Louis,Missouri,in1991. HereceivedanA.M.degreeinmathematicsfromBrownUniversity
in Providence, Rhode Island, in 1996; his Master’s thesis is titled Dimension Polynomials:
ComputationandApplications. HebeganworkintheDepartmentofMathematicsatNorth
Carolina State University in August, 1996. His research interests include differential Galois
theory, algebraic groups, and symbolic computation.
iii
Acknowledgements
I would like to express my sincerest gratitude to the following people and groups:
My advisor, Michael Singer, for giving me great ideas and tools for research; for sup-
porting me on numerous research assistantships and conference visits; and for showing
enthusiasm, patience and understanding throughout the whole process.
DanielBertrand,forhisroleindevelopingthematerialthatappearsinChapter3ofthis
dissertation. In particular, the new algorithm that appears in Section 3.4 was inspired by a
suggestion given to me by Professor Bertrand.
Marius van der Put, for inspiring and helping me with the material that appears in
Chapter 4.
Mohan Putcha, for helping with the material on semisimple groups in Section 4.4.
Ron Fulp, Kailash Misra and Larry Norris, for serving on my committee.
The members of the Symbolic Solutions Group at North Carolina State University and
the Kolchin Seminar in Differential Algebra at City College of New York for their encour-
agement and support.
My family and friends, for their love, support and patience which were essential to this
project.
iv
Contents
List of Tables vii
1 Introduction 1
2 Basic definitions and facts 3
2.1 Notation and results from linear algebra . . . . . . . . . . . . . . . . . . . . . 3
2.2 Algebraic groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.3 Differential algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3 Computing the group of L1◦L2, L1,L2 completely reducible 11
3.1 Connections and D-Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.2 Computing the group of L(y)=b, L completely reducible . . . . . . . . . . . 18
3.3 Computing the group of L1◦L2, L1,L2 completely reducible . . . . . . . . . 27
3.4 Computing the group of Y(cid:48) =AY +B, Y(cid:48) =AY completely reducible . . . . 41
4 Computing the group of D3+aD+b, a,b∈C[x] 53
4.1 Definitions and main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.2 Tori embedded in SL3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.3 Unipotent subgroups of SL3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.4 Semisimple subgroups of SL3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.5 Admissible subgroups of SL3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.6 Computing the group of D3+aD+b, a,b∈C[x] . . . . . . . . . . . . . . . . 89
4.7 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
5 Bibliography 119
v
A Maple code, documentation 125
A.1 README file . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
A.2 Maple code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
vi
List of Tables
4.1 Admissible subgroups of SL3. . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
vii
viii
Chapter 1
Introduction
A differential field is a field k equipped with a specified derivation operator. The subfield
C =C ofconstantsink isassumedtobealgebraicallyclosedandofcharacteristiczero. An
k
equation L(y) = 0 corresponds to an element L of the ring of differential operators k[D],
whichisnoncommutativeingeneral. ThereexistsaPicard-VessiotextensionK /kofdiffer-
L
ential fields, generated over k by the solution space V of L; it is the analogue of a splitting
L
field extension in polynomial Galois theory. The group G is the group of differential field
L
automorphisms of K /k. This group has a faithful representation as an algebraic subgroup
L
of GL(V ); thus, the groups in this case are linear algebraic groups. There is a full Galois
L
correspondence in this setting. See [Mag94] for an extensive introduction to differential
Galois theory. Chapter 2 of this dissertation presents basic results and notation from linear
algebra, algebraic groups and differential algebra.
As of now, there does not exist an algorithm to compute the group of a differential
operatorLingeneral. Thereexistalgorithmstocomputethegroupincertainspecialcases.
There also exist algorithms to perform related tasks. These related tasks include factoring
operators, computing associated operators, and testing for rational solutions. See [Sin99]
for an overview of these algorithms.
[CS99] presents a decision procedure to compute the group in case L is completely
reducible or, equivalently, G is a reductive group. [BS99] shows how to reduce the case of
a product of two completely reducible operators to that of a single completely reducible
operator; this article relies on results from cohomology presented in [Ber90] and [Ber92].
Chapter 3 of this dissertation presents the results from [BS99] and an optimization of the