Discovering Mathematics with Magma: Reducing the Abstract to the Concrete

386 Pages·2006·12.61 MB·English

by  Wieb Bosma

Algorithms and Computation
in Mathematics Volume 19
•
Editors
ArjehM.Cohen HenriCohen
DavidEisenbud BerndSturmfels
Wieb Bosma
John Cannon (Eds.)
Discovering Mathematics
with Magma
Reducing the Abstract to the Concrete
With9Figuresand16Tables
ABC
Editors
Dr.WiebBosma ProfessorDr.JohnCannon
KatholiekeUniv.Nijmegen UniversitySydney
VakgroepWiskunde DepartmentPureMathematicsF07
Postbus9010,6500GL 2006Sydney,Australia
Nijmegen,Netherlands E-mail:[email protected]
E-mail:[email protected]
LibraryofCongressControlNumber:2006931477
Mathematics Subject Classification (2000): 68W30, 11Yxx, 12Y05, 13Pxx, 14Qxx,
16Z05
ISSN1431-1550
ISBN-10 3-540-37632-1SpringerBerlinHeidelbergNewYork
ISBN-13 978-3-540-37632-3SpringerBerlinHeidelbergNewYork
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Preface
The appearance of this volume celebrates the first decade of Magma, a new
computeralgebrasystemlaunchedattheFirstMagmaConferenceonCompu-
tational Algebra held at Queen Mary and Westfield College, London, August
1993.
This book introduces the reader to the role Magma plays in advanced
mathematical research. Each paper examines how the computer can be used
to gain insight into either a single problem or a small group of closely related
problems. The intention is to present sufficient detail so that a reader can
(a), gain insight into the mathematical questions that are the origin of the
problems,and(b),developanunderstandingastohowsuchcomputationsare
specifiedinMagma.Itishopedthatthereaderwillcometoarealisationofthe
importantrolethatcomputationalalgebracanplayinmathematicalresearch.
Readers not primarily interested in using Magma will easily acquire the skills
neededtoundertakebasicprogramminginMagma,whileexperiencedMagma
users can learn both mathematics and advanced computational methods in
areas related to their own.
The core of the volume comprises 14 papers. The authors were invited to
submit articles on designated topics and these articles were then reviewed by
referees. Although by no means exhaustive, the topics range over a consider-
ablepartofMagma’scoverageofalgorithmicalgebra:fromnumbertheoryand
algebraicgeometry,viarepresentationtheoryandcomputationalgrouptheory
to some branches of discrete mathematics and graph theory. The papers are
preceded by an outline of the Magma project, a brief summary of the papers
and some instructions on reading the Magma code. A basic introduction to
the Magma language is given in an appendix.
Theeditorsexpresstheirgratitudetothecontributorstothisvolume,both
for the work put into producing the papers and for their patience.
John Cannon, Sydney
Wieb Bosma, Nijmegen
April 2006
Magma: The project
Computer Algebra
Computers are transforming the manner in which mathematics is discovered,
communicated and applied. Thus, computers are used todetermine the prop-
erties of, and relations between, mathematical objects, thereby enabling the
mathematician to formulate new conjectures or disprove old conjectures. The
application of mathematics often requires extensive calculations which are
practicalonlyifperformedbyacomputer.Increasingly,computersareusedto
generateorverifyformalmathematicalproofs,sometimesincombinationwith
extensive computations. Finally, computers allow mathematicians to typeset
their own papers and books, which can then be placed in a web archive for
instant access by colleagues in every part of the world.
The non-trivial use of computers in the creation and application of math-
ematics typically requires sophisticated software and it is here that a ma-
jor problem appears. Even experiments in elementary number theory require
multi-precision integer arithmetic, something which is found in very few gen-
eralscientificprogramminglanguages.Ifthesoftwarenecessaryforperforming
an algebraic calculation in an abstract area of mathematics has to be imple-
mented in a standard scientific programming language, it is likely to be at a
degreeofsophisticationandcomplexityfarbeyondtheabilityortime-budget
of most mathematicians.
General systems
Toaddressthisproblem,anumberofso-calledgeneralsymbolicComputerAl-
gebrasystemshaveemerged.Thesepackagesaredesignedfortheeasymanip-
ulation of symbolic expressions and have, as their central motivation, the so-
lutionofdifferentialequations.ThepackagesMACSYMAandReduce,whose
development commenced in the 1960’s, are two early examples. Currently,
Maple and Mathematica are the two most popular general symbolic systems.
VIII Magma: The project
The developers of these systems claim that they are suitable for all areas of
mathematics. However, while their symbolic-expression, transformation-rule
(SETR) model of computation with mathematical objects works well for the
large number of applied and theoretical problems that use tools such as cal-
culus and differential equations, it is inappropriate for much of mathematics.
For example, implementing code to determine the structure of a finite group
ineitherMapleorMathematicaisunnatural,difficultandresultsinextremely
inefficient code.
Specific area systems
The failure of general systems to meet the needs of many areas led to the
proliferationofspecialisedlibrariesandpackagesthatfocusonafairlynarrow
area.Inparticular,suchsystemscanprovidefunctionsthatarehighlyefficient
(compared to a general symbolic system) and which also provide specialised
coverage ofthearea.Furthermore,beingcomparatively smallmeansthatone
person or a small group can write and maintain the package. We will refer to
this class of packages as the specific area systems. The computational model
is usually quite different to the SETR model and it is often not explicit. An
important example of a specific area system is the number theory package
Pari,whichprovidesaClibraryoffunctionsforhigher-orderobjects(suchas
number fields and elliptic curves). Pari also provides a simple user interface
(GP) so that for many applications it is unnecessary to resort to writing C
code in order to use the package.
Thenarrowfocusofspecificareasystemshasamajordownsideinthatthe
user soon reaches the point where tools outside the scope of the package are
needed.Forexample,advancedmethodsforcomputingtheGaloisgroupofan
equation(anumbertheoryproblem)requiretheuseofsophisticatedtoolsfrom
computational group theory. This results in the user having to acquire and
learn a number of different packages and then somehow enable the packages
to communicate with one another. Generally speaking, mathematicians are
reluctant to invest the considerable time needed to gain a basic fluency with
more than one or two packages such as Pari.
The Magma project
A good computational model properly implemented should make it possi-
ble to code calculations at a high degree of abstraction in a manner which
is both concise and natural to the subject matter. Because of the quite dif-
ferent flavours of different parts of mathematics, it seems very unlikely that
a single model will work effectively across most of mathematics. A key aim
of the Magma project is the development of an effective model for a broad
area sharing a strong commonality of approach. We concentrate on topics
where the concepts of structure and morphism are central, areas where the
Magma: The project IX
SETR model provides an inadequate description of the underlying data and
itsoperations.Specifically,wechosetoconcentrateontopicsthatmakeheavy
use of the axiomatic approach to modern algebra as exemplified by van der
Waerden. This is taken to include algebra, number theory, algebraic geome-
try, algebraic topology and algebraic combinatorics. Indeed, modern algebra
provides a natural setting for the application of the Object-Oriented (OO)
programming paradigm, which is also the basis of the Axiom system.
Some general characteristics of Magma include:
• Concepts from universal algebra and category theory are used to provide
a formal basis for the computational model.
• The language design attempts to approximate the usual algebraic modes
ofthoughtandnotation.Inparticular,theprincipalconstructsintheuser
language are set, algebraic structure and morphism.
• Thesystemisdesignedtoautomaticallyrememberandutiliserelationships
between objects.
• Animportantgoalistoprovidehighly-tunedimplementationsofthemost
efficientalgorithmsknownforcomputingwiththevariousclassesofstruc-
ture supported. The objective is to provide performance similar to, or
better than, specialised stand-alone programs.
The computational model
Each object definable in Magma is considered to belong to a unique mathe-
maticalstructure(magma),calleditsparent,andthisparentmustbepresent
at the time the object is created. For instance, Magma does not have the
concept of a matrix as a primitive object; rather, a matrix may only exist as
an element of some particular ring or bimodule. Moreover, Magma employs
evaluation semantics, so that an expression may only be evaluated if every
identifier appearing in it has a value. Thus, notions such as “indeterminates”
or “unknowns” must be defined as elements of the appropriate magma, such
as a univariate polynomial ring over a field, before any use is made of them.
This philosophy makes programming a great deal easier, since there is never
any ambiguity as to the current interpretation of an object. In programming
language parlance, Magma is a strongly typed language.
Magmas are organised hierarchically. At the top level are varieties, which
arefamiliesofmagmassharingthesamebasicoperationsandaxioms.Typical
varieties include groups, commutative rings and modules. A variety has at-
tached to it functions (or methods in OO parlance) that do not depend upon
properties of a specific representation of a magma. When computing with
a structure A it is necessary that A be represented in some concrete form;
here we come to the level of categories. For example, within the variety of
groups there are categories for finitely-presented groups, permutation groups
and matrix groups (among others). The category to which a magma belongs
determinesnotonlyhowthemagmaanditselementsarerepresented,butalso
X Magma: The project
the permitted operations. Note that operations independent of the represen-
tation are inherited from the parent variety while representation-dependent
operations are attached to the category. Categories of magma that are spe-
cialised subcategories of an existing category C may, in turn, inherit opera-
tions from C, in addition to having further operations that are specific to the
subcategory.
Thegeneralmodelforconstructingamagmaistostartwitha‘free’magma
and then successively form submagmas, quotients and extensions until the
desired magma is obtained. This approach, which has its foundations in Uni-
versalAlgebra,providesapowerfulandgeneralmechanismforconstructinga
vastrangeofalgebraicandgeometricstructures.Sincemostcommonlyoccur-
ring structures in computational algebra are finitely generated, in general, a
magmaM isgivenintermsofafinitegeneratingset.Specialhandlingapplies
to the rare cases of infinitely generated magmas, such as R, C, and Laurent
series rings.
The user language
TheMagmauserlanguageisaninterpretedhigh-levellanguagewithdynamic
typing. The data structures are designed around the ideas of set, sequence,
mapping and structure. While the language formally belongs to the class of
imperative languages there is also a strong functional flavour. Powerful con-
structors allow complex sets and magmas to be constructed in a single state-
ment. Typically, a magma is created from another magma M by means of
constructor expressions, such as the submagma and quotient constructors:
sub< M | generators>
quo< M | normal subgroup>
These constructors not only return the new submagmaSor quotient Q, but
also return a morphism relating the new magma to the existing magma M
givenintheleftpartoftheconstructor;herethemorphismsaretheinclusion
monomorphism S→M and the natural epimorphism M→Q. Other map-
pingsfrommagmaM toM maybedirectlycreatedbymeansofthemapping
1 2
or the homomorphism constructors:
map< M → M | x (cid:2)→ expressioninx >
1 2
hom< M → M | imagesof generatorsof M >
1 2 1
Magma has a powerful collection of set and sequence constructors whose syn-
tax is based on traditional mathematical set notation and was influenced by
SETL [1]. For example, {1..10} is the set of integers {x∈Z|1≤x≤10} and
{ i2 : i in{1..10} | not IsPrime(i) };
{ 1, 16, 36, 64, 81, 100 }
is the set of squares of the non-primes in this set.
Magma: The project XI
The language has a functional subset, providing functions and procedures
asfirst-classobjects.Functionsareallowedmultiplereturnvalues.Valueargu-
ments and parameters are supported both for functions and procedures while
reference arguments are supported for procedures only. As first-class objects,
functionsandproceduresarecreatedasthevaluesofexpressions,ratherthan
by means of special statements.
For a more complete description, the reader should consult the concise
introductionintheappendixonpage331,or[2]and[3]foraformaloverview,
or [4] and [5] for full user documentation.
Acknowledgements
The core of the Magma system has been developed by the Computational
Algebra Group at the University of Sydney. In addition, Magma includes
algorithms and code generously contributed by many mathematicians from
around the world. In many ways theMagma project represents a major coop-
erative effort by mathematicians to develop a system properly tuned to their
needs. A full list of contributors may be found at the front of Volume 1 of [5].
References
1. J.T.Schwartz,R.B.K.Dewar,E.Dubinsky,E.Schonberg, Programming With
Sets: An Introduction to SETL, New York: Springer, 1986.
2. WiebBosma,JohnCannon,CatherinePlayoust, TheMagmaalgebrasystemI:
The user language, J. Symbolic Comput. 24 (1997), 235–265.
See also the Magma home page at http://magma.maths.usyd.edu.au/magma/.
3. Wieb Bosma, John Cannon, Graham Matthews, Programming with algebraic
structures:designoftheMagmalanguage,pp.52–57in:M.Giesbrecht(ed),Pro-
ceedings of International Symposium on Symbolic and Algebraic Computation,
Oxford (1994), Association for Computing Machinery, 1994.
4. Wieb Bosma, John Cannon, Catherine Playoust, Algebraic programming with
Magma, to appear.
5. JohnCannon,WiebBosma(eds.),HandbookofMagmaFunctions,Version2.13,
Volumes 1–10, Sydney, July 2006, 4320 pages.
Discovering mathematics: About this volume
Introduction
A major motivation behind the design and implementation of Magma has
been the desire to make advanced areas of algebra and geometry much more
accessible to effective computation. We believe that the system represents
a major step towards achieving these aims. Since its first public release in
1994, Magma has been widely adopted and has been used for applications in
a wide variety of fields both inside and outside of mathematics. Magma has
been used by many mathematicians to develop packages for new areas. Ex-
amplesincludealgebraicgeometryviaschemes(GavinBrown),cohomological
rings of group modules (Jon Carlson), quantum groups (Willem de Graaf),
Kronecker’s algorithm for solving systems of polynomial equations (Gregoire
Lecerf),Aschbacheranalysisoflargedegreematrixgroups(EamonnO’Brien),
hyperelliptic curves (Michael Stoll) and modular forms and modular abelian
varieties (William Stein). Magma has been cited in more than 2000 books,
papers and theses.
ThisvolumeisacollectionofcasestudiesshowinghowMagmaisappliedto
problemsacrossarangeofdifferentareas.Inmostinstancestheworkledtothe
publicationofoneormoretheoreticalpapers.Thecorrespondingpaperinthis
volume presentsthe Magma computations behind the theoretical results.The
collection of papers can be regarded as a series of tutorials on the application
of Magma to advanced research problems. The authors were asked to provide
some mathematical background on their problem(s) and then present the
Magmacodewithanextendedcommentary.Itishopedthatreaderswillgain
insight into the possible applications of Magma and also detailed examples
of its use in various contexts. For the convenience of readers not particularly
familiarwiththeMagmalanguage,abriefintroductionwrittenbyGeoffBailey
isincludedasanappendixtothisbook.Inaddition,areadermeetingMagma
forthefirsttimemayalsowishtoconsultVolume1oftheHandbookofMagma
Functions (shipped in electronic form as part of the Magma distribution) in
order to get a detailed description of the language and its intrinsic functions.