Finite Geometry and Combinatorial Applications

297 Pages·2015·1.866 MB·English

by  Simeon Ball

LONDONMATHEMATICALSOCIETYSTUDENTTEXTS
ManagingEditor:ProfessorD.Benson,
DepartmentofMathematics,UniversityofAberdeen,UK
42 Equilibriumstatesinergodictheory,GERHARDKELLER
43 Fourieranalysisonfinitegroupsandapplications,AUDREYTERRAS
44 Classicalinvarianttheory,PETERJ.OLVER
45 Permutationgroups,PETERJ.CAMERON
47 Introductorylecturesonringsandmodules.JOHNA.BEACHY
48 Settheory,ANDRÁSHAJNAL&PETERHAMBURGER.TranslatedbyATTILAMATE
49 AnintroductiontoK-theoryforC*-algebras,M.RØRDAM,F.LARSEN&N.J.
LAUSTSEN
50 Abriefguidetoalgebraicnumbertheory,H.P.F.SWINNERTON-DYER
51 Stepsincommutativealgebra:Secondedition,R.Y.SHARP
52 FiniteMarkovchainsandalgorithmicapplications,OLLEHÄGGSTRÖM
53 Theprimenumbertheorem,G.J.O.JAMESON
54 Topicsingraphautomorphismsandreconstruction,JOSEFLAURI&RAFFAELE
SCAPELLATO
55 Elementarynumbertheory,grouptheoryandRamanujangraphs,GIULIANADAVIDOFF,
PETERSARNAK&ALAINVALETTE
56 Logic,inductionandsets,THOMASFORSTER
57 IntroductiontoBanachalgebras,operatorsandharmonicanalysis,GARTHDALESetal.
58 Computationalalgebraicgeometry,HALSCHENCK
59 Frobeniusalgebrasand2-Dtopologicalquantumfieldtheories,JOACHIMKOCK
60 Linearoperatorsandlinearsystems,JONATHANR.PARTINGTON
61 AnintroductiontononcommutativeNoetherianrings:Secondedition,K.R.GOODEARL&
R.B.WARFIELD,JR
62 Topicsfromone-dimensionaldynamics,KARENM.BRUCKS&HENKBRUIN
63 Singularpointsofplanecurves,C.T.C.WALL
64 AshortcourseonBanachspacetheory,N.L.CAROTHERS
65 ElementsoftherepresentationtheoryofassociativealgebrasI,IBRAHIMASSEM,DANIEL
SIMSON&ANDRZEJSKOWRON´SKI
66 Anintroductiontosievemethodsandtheirapplications,ALINACARMENCOJOCARU&
M.RAMMURTY
67 Ellipticfunctions,J.V.ARMITAGE&W.F.EBERLEIN
68 Hyperbolicgeometryfromalocalviewpoint,LINDAKEEN&NIKOLALAKIC
69 LecturesonKählergeometry,ANDREIMOROIANU
70 Dependencelogic,JOUKUVÄÄNÄNEN
71 ElementsoftherepresentationtheoryofassociativealgebrasII,DANIELSIMSON&
ANDRZEJSKOWRON´SKI
72 ElementsoftherepresentationtheoryofassociativealgebrasIII,DANIELSIMSON&
ANDRZEJSKOWRON´SKI
73 Groups,graphsandtrees,JOHNMEIER
74 RepresentationtheoremsinHardyspaces,JAVADMASHREGHI
75 Anintroductiontothetheoryofgraphspectra,DRAGOŠCVETKOVIC´,PETER
ROWLINSON&SLOBODANSIMIC´
76 NumbertheoryinthespiritofLiouville,KENNETHS.WILLIAMS
77 Lecturesonprofinitetopicsingrouptheory,BENJAMINKLOPSCH,NIKOLAYNIKOLOV
&CHRISTOPHERVOLL
78 Cliffordalgebras:anintroduction,D.J.H.GARLING
79 IntroductiontocompactRiemannsurfacesanddessinsd’enfants,ERNESTOGIRONDO&
GABINOGONZÁLEZ-DIEZ
80 TheRiemannhypothesisforfunctionfields,MACHIELVANFRANKENHUIJSEN
81 Numbertheory,Fourieranalysisandgeometricdiscrepancy,GIANCARLOTRAVAGLINI
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CBO9781316257449
LondonMathematicalSocietyStudentTexts82
Finite Geometry and Combinatorial
Applications
SIMEON BALL
UniversitatPolitècnicadeCatalunya,Barcelona
21:42:39 BST 2016.
CBO9781316257449
UniversityPrintingHouse,CambridgeCB28BS,UnitedKingdom
CambridgeUniversityPressispartoftheUniversityofCambridge.
ItfurtherstheUniversity’smissionbydisseminatingknowledgeinthepursuitof
education,learningandresearchatthehighestinternationallevelsofexcellence.
www.cambridge.org
Informationonthistitle:www.cambridge.org/9781107107991
©SimeonBall2015
Thispublicationisincopyright.Subjecttostatutoryexception
andtotheprovisionsofrelevantcollectivelicensingagreements,
noreproductionofanypartmaytakeplacewithoutthewritten
permissionofCambridgeUniversityPress.
Firstpublished2015
PrintedintheUnitedStates of America by Sheridan Books, Inc.
AcataloguerecordforthispublicationisavailablefromtheBritishLibrary
LibraryofCongressCataloguinginPublicationdata
Ball,Simeon(SimeonMichael)
Finitegeometryandcombinatorialapplications/SimeonBall,
UniversitatPolitècnicadeCatalunya,Barcelona.
pages cm.–(LondonMathematicalSocietystudenttexts;82)
Includesbibliographicalreferencesandindex.
ISBN978-1-107-10799-1(Hardback:alk.paper)–
ISBN978-1-107-51843-8(Paperback:alk.paper)
1. Finitegeometries. 2. Combinatorialanalysis. I. Title.
QA167.2.B352015
516(cid:2).11–dc23 2015009563
ISBN978-1-107-10799-1Hardback
ISBN978-1-107-51843-8Paperback
CambridgeUniversityPresshasnoresponsibilityforthepersistenceoraccuracyof
URLsforexternalorthird-partyinternetwebsitesreferredtointhispublication,
anddoesnotguaranteethatanycontentonsuchwebsitesis,orwillremain,
accurateorappropriate.
21:42:39 BST 2016.
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Contents
Preface pageix
Notation xi
1 Fields 1
1.1 Ringsandfields 1
1.2 Fieldautomorphisms 6
1.3 Themultiplicativegroupofafinitefield 9
1.4 Exercises 10
2 Vectorspaces 15
2.1 Vectorspacesandsubspaces 15
2.2 Linearmapsandlinearforms 17
2.3 Determinants 19
2.4 Quotientspaces 20
2.5 Exercises 21
3 Forms 25
3.1 σ-Sesquilinearforms 25
3.2 Classificationofreflexiveforms 27
3.3 Alternatingforms 30
3.4 Hermitianforms 34
3.5 Symmetricforms 38
3.6 Quadraticforms 40
3.7 Exercises 47
4 Geometries 51
4.1 Projectivespaces 51
4.2 Polarspaces 54
4.3 Quotientgeometries 60
v
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vi Contents
4.4 Countingsubspaces 61
4.5 Generalisedpolygons 65
4.6 Plückercoordinates 71
4.7 Polarities 74
4.8 Ovoids 76
4.9 Exercises 83
5 Combinatorialapplications 93
5.1 Groups 93
5.2 Finiteanaloguesofstructuresinrealspace 99
5.3 Codes 105
5.4 Graphs 109
5.5 Designs 114
5.6 Permutationpolynomials 117
5.7 Exercises 120
6 Theforbiddensubgraphproblem 124
6.1 TheErdo˝s–Stonetheorem 124
6.2 Evencycles 125
6.3 Completebipartitegraphs 130
6.4 GraphscontainingnoK 132
2,s
6.5 AprobabilisticconstructionofgraphscontainingnoK 134
t,s
6.6 GraphscontainingnoK 135
3,3
6.7 Thenormgraph 137
6.8 GraphscontainingnoK 140
5,5
6.9 Exercises 144
7 MDScodes 147
7.1 Singletonbound 147
7.2 LinearMDScodes 148
7.3 DualMDScodes 151
7.4 TheMDSconjecture 152
7.5 Polynomialinterpolation 154
7.6 TheA-functions 155
7.7 Lemmaoftangents 157
7.8 Combininginterpolationwiththelemmaoftangents 162
7.9 AproofoftheMDSconjecturefork(cid:2)p 164
7.10 MoreexamplesofMDScodesoflengthq+1 165
7.11 ClassificationoflinearMDScodesoflengthq+1for
k(cid:2)p 167
7.12 ThesetoflinearformsassociatedwithalinearMDScode 172
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Contents vii
7.13 Lemmaoftangentsinthedualspace 174
7.14 The algebraic hypersurface associated with a linear
MDScode 177
7.15 ExtendabilityoflinearMDScodes 182
7.16 ClassificationoflinearMDScodesoflengthq+1for
√
k<c q 184
√
7.17 AproofoftheMDSconjecturefork<c q 189
7.18 Exercises 189
AppendixA Solutionstotheexercises 191
A.1 Fields 191
A.2 Vectorspaces 200
A.3 Forms 206
A.4 Geometries 213
A.5 Combinatorialapplications 229
A.6 Theforbiddensubgraphproblem 233
A.7 MDScodes 238
AppendixB Additionalproofs 242
B.1 Probability 242
B.2 Fields 243
B.3 Commutativealgebra 247
AppendixC Notesandreferences 263
C.1 Fields 263
C.2 Vectorspaces 264
C.3 Forms 264
C.4 Geometries 264
C.5 Combinatorialapplications 266
C.6 Theforbiddensubgraphproblem 269
C.7 MDScodes 270
C.8 Appendices 271
References 272
Index 282
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Preface
This book is essentially a text book that introduces the geometrical objects
whichariseinthestudyofvectorspacesoverfinitefields.Itadvancesrapidly
throughthebasicmaterial,enablingthereadertoconsiderthemoreinteresting
aspects of the subject without having to labour excessively. There are over a
hundredexerciseswhichcontainalotofcontentnotincludedinthetext.This
shouldbetakenintoconsiderationandeventhoughonemaynotwishtotryto
solvetheexercisesthemselves,theyshouldnotbeignored.Therearedetailed
solutionsprovidedtoalltheexercises.
Thefirstfourchapterstreatthealgebraicandgeometricaspectsoffinitevec-
torspaces.Thefollowingthreechaptersconsistofcombinatorialapplications.
Thereisachaptercontainingabrieftreatmentofapplicationstogroups,real
geometry,codes,graphs,designsandpermutationpolynomials.Thenthereisa
chapterthatgivesamorein-depthtreatmentofapplicationstoextremalgraph
theory, specifically the forbidden subgraph problem, and then a chapter on
maximumdistanceseparablecodes.
This book is self-contained in the sense that any theorem or lemma which
issubsequentlyusedisproven.TheonlyexceptionstothisareBombieri’sthe-
oremandtheHuxely–Iwaniectheoremconcerningthedistributionofprimes,
whichareusedinthechapterontheforbiddensubgraphproblem,theHasse–
Weiltheorem,whichisusedtoboundthenumberofpointsonaplanealgebraic
curve at the end of the chapter on maximum distance separable codes, and
Hilbert’sNullstellensatz,whichisusedintheappendixoncommutativealge-
bra. Although there are almost no prerequisites, it would be helpful to have
studiedpreviouslysomebasicalgebraandlinearalgebra,sinceotherwisethe
firstcoupleofchaptersmayappearsomewhatbrief.Therearesometheorems
thatarequotedwithoutproof,butinallcasestheseappearattheendofsome
branchandarenotbuiltupon.Therearesometheoremswhoseproofappears
ix
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x Preface
inAppendixB.Thisisdonewhentheproofofsomeparticulartheoremmay
interrupttheflowofthebook.
Howtousethisbookif...
...you are not teaching a course. For many readers a lot of the material in
Chapter1andChapter2willbefamiliar.However,someoftheexercises,those
relatingtolatinsquares,semifieldsandspreads,maynotbeandare,although
notgenerallyessential,atleastrelevanttowhatappearsinlaterchapters.For
this reason they should not be overlooked. There is no need to read all the
detailsofChapter3.ItisenoughtoreadasfarasTheorem3.6,chooseoneof
theσ-sesquilinearformstoconsiderinmoredetailandSection3.6.Thecentral
chaptersofthebookareChapter4andChapter5.
...you are teaching a course. This book is not structured as lecture notes.
However, there is plenty of material to plan a course, even within a pre-
established syllabus. Note that a lot of the material is contained in exercises
that, since the solutions are provided, can be explained as theorems in class.
Onecouldteachthefollowingcourse.
(1) Latinsquares.DefinitionandexercisesfromChapter1andusetheselec-
turesto(re-)introducethestudenttofinitefields.
(2) Affineplanes.ExercisesinChapter4,usesomeastheoremsandleavethe
restasexercises.
(3) Projectiveplanes.TextandexercisesinChapter4,introducingexampleof
PG (F )andDesargues’theorem.
2 q
(4) Projectivespaces.UseChapter4.
(5) Polarspaces.Sketchclassificationofσ-sesquilinearforms,i.e.Chapter3
asfarasTheorem3.6andsketchSection3.6.ThenTheorem4.3.
(6) Quotientspaces.Section4.3andSection4.4.
(7) Generalisedpolygons.Section4.5.
(8) Ovalsandovoids.Section4.8andincludeSegre’stheorem,Theorem4.38.
One could then pick and choose from Chapter 5 and maybe Chapter 6.
Although it may be disheartening to see a full set of solutions, many of the
exercises can be easily adapted so that exercise sheets, which do not have
solutions,canbecompiledifnecessary.
BynomeansdoIconsiderthecontentsofthisbooktobeanunbiasedview
ofwhatfinitegeometryis.ThereareaspectsofthesubjectthatIhavebarely
toucheduponandsomeIhavenotmentionedatall.Ihavestuck,inthemain
part,tothatwhichisofinteresttomeandthatIfeelconfidentenoughtowrite
about.
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