BERMAN, PETER HILLEL. Computing Galois Groups for Certain Classes of Ordinary

Di¬erential Equations. (Under the direction of Michael Singer.)

As of now, it is an open problem to ¬nd an algorithm that computes the Galois group

G of an arbitrary linear ordinary di¬erential operator L ∈ C(x)[D]. We assume that C

is a computable, characteristic-zero, algebraically closed constant ¬eld with factorization

algorithm. In this dissertation, we present new methods for computing di¬erential 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 di¬erential 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 ¬eld 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 ful¬llment 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, in 1991. He received an A.M. degree in mathematics from Brown University

in Providence, Rhode Island, in 1996; his Master™s thesis is titled Dimension Polynomials:

Computation and Applications. He began work in the Department of Mathematics at North

Carolina State University in August, 1996. His research interests include di¬erential 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.

Daniel Bertrand, for his role in developing the material that appears in Chapter 3 of this

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 Di¬erential 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 de¬nitions and facts 3

2.1 Notation and results from linear algebra . . . . . . . . . . . . . . . . . . . . . 3

2.2 Algebraic groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.3 Di¬erential algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3 Computing the group of L1 —¦ L2 , L1 , L2 completely reducible 11

Connections and D-Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.1

3.2 Computing the group of L(y) = b, L completely reducible . . . . . . . . . . . 18

Computing the group of L1 —¦ L2 , L1 , L2 completely reducible . . . . . . . . . 27

3.3

3.4 Computing the group of Y = AY + B, Y = AY completely reducible . . . . 41

4 Computing the group of D3 + aD + b, a, b ∈ C[x] 53

4.1 De¬nitions and main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

Tori embedded in SL3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.2

Unipotent subgroups of SL3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.3

Semisimple subgroups of SL3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.4

Admissible subgroups of SL3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.5

Computing the group of D3 + aD + b, a, b ∈ C[x] . . . . . . . . . . . . . . . . 89

4.6

4.7 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

5 Bibliography 119

v

A Maple code, documentation 125

A.1 README ¬le . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

A.2 Maple code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

vi

List of Tables

Admissible subgroups of SL3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.1

vii

viii

Chapter 1

Introduction

A di¬erential ¬eld is a ¬eld k equipped with a speci¬ed derivation operator. The sub¬eld

C = Ck of constants in k is assumed to be algebraically closed and of characteristic zero. An

equation L(y) = 0 corresponds to an element L of the ring of di¬erential operators k[D],

which is noncommutative in general. There exists a Picard-Vessiot extension KL /k of di¬er-

ential ¬elds, generated over k by the solution space VL of L; it is the analogue of a splitting

¬eld extension in polynomial Galois theory. The group GL is the group of di¬erential ¬eld

automorphisms of KL /k. This group has a faithful representation as an algebraic subgroup

of GL(VL ); thus, the groups in this case are linear algebraic groups. There is a full Galois

correspondence in this setting. See [Mag94] for an extensive introduction to di¬erential

Galois theory. Chapter 2 of this dissertation presents basic results and notation from linear

algebra, algebraic groups and di¬erential algebra.

As of now, there does not exist an algorithm to compute the group of a di¬erential

operator L in general. There exist algorithms to compute the group in certain special cases.

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

core decision procedure.

Chapter 3 is organized as follows: Section 3.1 discusses connections and D-modules and

their relation to di¬erential equations and systems. Section 3.2 presents the results from

Section 2 of [BS99]. This section includes Algorithm I, which computes the group of the

inhomogeneous equation L(y) = b, L ∈ C(x)[D] completely reducible, b ∈ C(x). Section 3.3

presents the results from Section 3 of [BS99]. This section includes Algorithm II, which

computes the group of L1 (L2 (y)) = 0, L1 , L2 ∈ C(x) completely reducible. Algorithm II

ˆ ˆ

works by computing an associated inhomogeneous equation L(y) = b, where L is completely

reducible, and applying part of Algorithm I to that equation. Section 3.4 presents Algo-

rithm III, an optimization of Algorithm I: Whereas Algorithm I relies on parameterizing

all factorizations of L to compute the group of L(y) = b, Algorithm III only requires a

single expression of L as the least common left multiple of a set of irreducible operators.

Note that Algorithm III is presented in terms of inhomogeneous ¬rst-order systems rather

than inhomogeneous equations; the results of Section 3.1 show that the two settings are

interchangeable in our case.

In Chapter 4, we give a set of criteria to compute the Galois group of a di¬erential

equation of the form y (3) + ay + by = 0, a, b ∈ C[x], where C is a computable, algebraically

closed constant ¬eld of characteristic zero. Moreover, we present an algorithm to carry out

¯

this computation in case C = Q, the ¬eld of algebraic numbers. This algorithm applies the

approach used in [dP98b] to study order-two equations with one or two singular points.

Our method relies on a result of Ramis that says that the group of such an equation

must be connected and have defect zero (c.f. [MS96] and [dP98a]). In Section 4.1, we state

this result; we also state the main theorem of Chapter 4. In sections 4.2, 4.3 and 4.4,

we enumerate the tori, unipotent groups, and semisimple groups that can be embedded in

SL3 (C) along with their conjugacy classes in SL3 (C). In Section 4.5, we use the results from

Sections 4.2-4.4 to produce a complete list of conjugacy classes of all subgroups of SL3 (C)

that occur as Galois groups of equations of the prescribed form. In Section 4.6, we give an

¯

algorithm to compute the group of an equation of this form, in the case where C = Q. Each

step of the algorithm employs a simple, implementable test. Examples of this algorithm are

given in Section 4.7. The author has implemented this algorithm in Maple. The code is

provided in an appendix.

2

Chapter 2

Basic de¬nitions and facts

2.1 Notation and results from linear algebra

This section is a review of certain facts from basic linear algebra (see, for instance, [HK71]),

included here to establish notation for the remainder of the document.

Let V (resp., W ) be an m- (resp., n-) dimensional vector space over a ¬eld k with ordered

m

basis E = {e1 , . . . , em } (resp., F = {f1 , . . . , fn }). For any vector v = vi ei ∈ V, we write

i=1

[v]E = (v1 , . . . , vm )T ∈ k m .

Given φ ∈ Homk (V, W ), suppose

n

aji fj , aji ∈ k.

φ(ei ) =

j=1

Then we de¬ne the matrix [φ]E,F by

[φ]E,F = (aij ) ∈ k n—m ;

this formula yields the matrix-by-vector multiplication formula

[φ(v)]F = [φ]E,F [v]E for all v ∈ V.

In the special case V = W (i.e., φ ∈ Endk (V )) and E = F, we write [φ]E in place of [φ]E,F . In

the special case V = W, φ = id, E = F, let P = [id]E,F ; we see that P is the change-of-basis

matrix from coordinates in E to coordinates in F.

3

The following facts are well-known: If X is a k-vector space with ordered basis B =

{b1 , b2 , . . . , bq } and ψ ∈ Homk (W, X), then