Noncommutative Involutive Bases [PhD thesis]

389 Pages·2006·2.621 MB·English

by  Gareth Alun Evans

Noncommutative Involutive Bases
Thesis submitted to the University of Wales in support of
the application for the degree of Philosophiæ Doctor
by
Gareth Alun Evans
School of Informatics
The University of Wales, Bangor
September 2005
DECLARATION
This work has not previously been accepted in substance for any degree and is not being
concurrently submitted in candidature for any degree.
Signed ....................................................... (candidate)
Date ........................................................
STATEMENT 1
This thesis is the result of my own investigations, except where otherwise stated.
Other sources are acknowledged by explicit references. A bibliography is appended.
Signed ....................................................... (candidate)
Date ........................................................
STATEMENT 2
I hereby give consent for my thesis, if accepted, to be available for photocopying and
for inter-library loan, and for the title and summary to be made available to outside
organisations.
Signed ....................................................... (candidate)
Date ........................................................
Summary
The theory of Gr¨obner Bases originated in the work of Buchberger [11] and is now con-
sidered to be one of the most important and useful areas of symbolic computation. A
great deal of effort has been put into improving Buchberger’s algorithm for computing a
Gr¨obner Basis, and indeed in finding alternative methods of computing Gr¨obner Bases.
Two of these methods include the Gr¨obner Walk method [1] and the computation of
Involutive Bases [58].
By the mid 1980’s, Buchberger’s work had been generalised for noncommutative poly-
nomial rings by Bergman [8] and Mora [45]. This thesis provides the corresponding
generalisation for Involutive Bases and (to a lesser extent) the Gr¨obner Walk, with the
main results being as follows.
(1) Algorithms for several new noncommutative involutive divisions are given, including
strong; weak; global and local divisions.
(2) An algorithm for computing a noncommutative Involutive Basis is given. When used
with one of the aforementioned involutive divisions, it is shown that this algorithm
returns a noncommutative Gr¨obner Basis on termination.
(3) An algorithm for a noncommutative Gr¨obner Walk is given, in the case of conversion
between two harmonious monomial orderings. It is shown that this algorithm gener-
alises to give an algorithm for performing a noncommutative Involutive Walk, again
in the case of conversion between two harmonious monomial orderings.
(4) Two new properties of commutative involutive divisions are introduced (stability and
extendibility), respectively ensuring the termination of the Involutive Basis algorithm
and the applicability (under certain conditions) of homogeneous methods of comput-
ing Involutive Bases.
Source code for an initial implementation of an algorithm to compute noncommutative
Involutive Bases is provided in Appendix B. This source code, written using ANSI C and
a series of libraries (AlgLib) provided by MSSRC [46], forms part of a larger collection of
programsprovidingexamplesforthethesis,includingimplementationsofthecommutative
and noncommutative Gr¨obner Basis algorithms [11, 45]; the commutative Involutive Basis
algorithm for the Pommaret and Janet involutive divisions [58]; and the Knuth-Bendix
critical pairs completion algorithm for monoid rewrite systems [39].
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Acknowledgements
Many people have inspired me to complete this thesis, and I would like to take this
opportunity to thank some of them now.
I would like to start by thanking my family for their constant support, especially my
parents who have encouraged me every step of the way. Mae fy nyled yn fawr iawn i chi.
I would like to thank Prof. Larry Lambe from MSSRC, whose software allowed me to
test my theories in a way that would not have been possible elsewhere.
Thanks to all the Mathematics Staff and Students I have had the pleasure of working with
over the past seven years. Particular thanks go to Dr. Bryn Davies, who encouraged me
to think independently; to Dr. Jan Abas, who inspired me to reach goals I never thought
I could reach; and to Prof. Ronnie Brown, who introduced me to Involutive Bases.
I would like to finish by thanking my Supervisor Dr. Chris Wensley. Our regular meetings
kept the cogs in motion and his insightful comments enabled me to avoid wrong turnings
and to get the little details right. Diolch yn fawr!
This work has been gratefully supported by the EPSRC and by the School of Informatics
at the University of Wales, Bangor.
Typeset using LATEX, XFig andXY-pic.
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“No one has ever done anything like this.”
“That’s why it’s going to work.”
The Matrix [54]
iv
Contents
Summary ii
Acknowledgements iii
Introduction 1
Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Structure and Principal Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1 Preliminaries 9
1.1 Rings and Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.1.1 Groups and Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.1.2 Polynomial Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.1.3 Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.2 Monomial Orderings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.2.1 Commutative Monomial Orderings . . . . . . . . . . . . . . . . . . 14
1.2.2 Noncommutative Monomial Orderings . . . . . . . . . . . . . . . . 16
1.2.3 Polynomial Division . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2 Commutative Gr¨obner Bases 22
2.1 S-polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.2 Dickson’s Lemma and Hilbert’s Basis Theorem . . . . . . . . . . . . . . . . 27
2.3 Buchberger’s Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
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2.4 Reduced Gr¨obner Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.5 Improvements to Buchberger’s Algorithm . . . . . . . . . . . . . . . . . . . 34
2.5.1 Buchberger’s Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.5.2 Homogeneous Gr¨obner Bases . . . . . . . . . . . . . . . . . . . . . . 39
2.5.3 Selection Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.5.4 Basis Conversion Algorithms . . . . . . . . . . . . . . . . . . . . . . 42
2.5.5 Optimal Variable Orderings . . . . . . . . . . . . . . . . . . . . . . 43
2.5.6 Logged Gr¨obner Bases . . . . . . . . . . . . . . . . . . . . . . . . . 43
3 Noncommutative Gr¨obner Bases 46
3.1 Overlaps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.2 Mora’s Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.2.1 Termination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.3 Reduced Gr¨obner Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.4 Improvements to Mora’s Algorithm . . . . . . . . . . . . . . . . . . . . . . 56
3.4.1 Buchberger’s Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.4.2 Homogeneous Gr¨obner Bases . . . . . . . . . . . . . . . . . . . . . . 59
3.4.3 Selection Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.4.4 Logged Gr¨obner Bases . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.5 A Worked Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.5.1 Initialisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.5.2 Calculating and Reducing S-polynomials . . . . . . . . . . . . . . . 64
3.5.3 Applying Buchberger’s Second Criterion . . . . . . . . . . . . . . . 66
3.5.4 Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4 Commutative Involutive Bases 70
4.1 Involutive Divisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.1.1 Involutive Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . 73
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4.1.2 Thomas, Pommaret and Janet divisions . . . . . . . . . . . . . . . . 74
4.2 Prolongations and Autoreduction . . . . . . . . . . . . . . . . . . . . . . . 77
4.3 Continuity and Constructivity . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.4 The Involutive Basis Algorithm . . . . . . . . . . . . . . . . . . . . . . . . 86
4.5 Improvements to the Involutive Basis Algorithm . . . . . . . . . . . . . . . 96
4.5.1 Improved Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . 96
4.5.2 Homogeneous Involutive Bases . . . . . . . . . . . . . . . . . . . . . 96
4.5.3 Logged Involutive Bases . . . . . . . . . . . . . . . . . . . . . . . . 102
5 Noncommutative Involutive Bases 104
5.1 Noncommutative Involutive Reduction . . . . . . . . . . . . . . . . . . . . 105
5.2 Prolongations and Autoreduction . . . . . . . . . . . . . . . . . . . . . . . 110
5.3 The Noncommutative Involutive Basis Algorithm . . . . . . . . . . . . . . 112
5.4 Continuity and Conclusivity . . . . . . . . . . . . . . . . . . . . . . . . . . 112
5.4.1 Properties for Strong Involutive Divisions . . . . . . . . . . . . . . . 114
5.4.2 Properties for Weak Involutive Divisions . . . . . . . . . . . . . . . 117
5.5 Noncommutative Involutive Divisions . . . . . . . . . . . . . . . . . . . . . 119
5.5.1 Two Global Divisions . . . . . . . . . . . . . . . . . . . . . . . . . . 119
5.5.2 An Overlap-Based Local Division . . . . . . . . . . . . . . . . . . . 124
5.5.3 A Strong Local Division . . . . . . . . . . . . . . . . . . . . . . . . 135
5.5.4 Alternative Divisions . . . . . . . . . . . . . . . . . . . . . . . . . . 142
5.6 Termination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
5.7 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
5.7.1 A Worked Example . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
5.7.2 Involutive Rewrite Systems . . . . . . . . . . . . . . . . . . . . . . 154
5.7.3 Comparison of Divisions . . . . . . . . . . . . . . . . . . . . . . . . 157
5.8 Improvements to the Noncommutative Involutive Basis Algorithm . . . . . 158
5.8.1 Efficient Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
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5.8.2 Improved Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . 161
5.8.3 Logged Involutive Bases . . . . . . . . . . . . . . . . . . . . . . . . 162
6 Gr¨obner Walks 165
6.1 Commutative Walks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
6.1.1 Matrix Orderings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
6.1.2 The Commutative Gr¨obner Walk Algorithm . . . . . . . . . . . . . 167
6.1.3 A Worked Example . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
6.1.4 The Commutative Involutive Walk Algorithm . . . . . . . . . . . . 176
6.2 Noncommutative Walks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
6.2.1 Functional Decompositions . . . . . . . . . . . . . . . . . . . . . . . 177
6.2.2 The Noncommutative Gr¨obner Walk Algorithm for Harmonious
Monomial Orderings . . . . . . . . . . . . . . . . . . . . . . . . . . 179
6.2.3 A Worked Example . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
6.2.4 The Noncommutative Involutive Walk Algorithm for Harmonious
Monomial Orderings . . . . . . . . . . . . . . . . . . . . . . . . . . 185
6.2.5 A Worked Example . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
6.2.6 Noncommutative Walks Between Any Two Monomial Orderings? . 190
7 Conclusions 191
7.1 Current State of Play . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
7.2 Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
A Proof of Propositions 5.5.31 and 5.5.32 194
A.1 Proposition 5.5.31 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
A.2 Proposition 5.5.32 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
B Source Code 212
B.1 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
B.1.1 MSSRC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
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B.1.2 AlgLib . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
B.2 Listings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
B.2.1 README . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
B.2.2 arithmetic functions.h . . . . . . . . . . . . . . . . . . . . . . . . . 219
B.2.3 arithmetic functions.c . . . . . . . . . . . . . . . . . . . . . . . . . 220
B.2.4 file functions.h . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226
B.2.5 file functions.c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
B.2.6 fralg functions.h . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238
B.2.7 fralg functions.c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240
B.2.8 list functions.h . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268
B.2.9 list functions.c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271
B.2.10 ncinv functions.h . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290
B.2.11 ncinv functions.c . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291
B.2.12 involutive.c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340
C Program Output 356
C.1 Sample Sessions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356
C.1.1 Session 1: Locally Involutive Bases . . . . . . . . . . . . . . . . . . 356
C.1.2 Session 2: Involutive Complete Rewrite Systems . . . . . . . . . . . 358
C.1.3 Session 3: Noncommutative Involutive Walks . . . . . . . . . . . . . 361
C.1.4 Session 4: Ideal Membership . . . . . . . . . . . . . . . . . . . . . . 365
Bibliography 368
Index 374
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