Unit Equations in Diophantine Number Theory

375 Pages·2015·2.46 MB·English

by  Jan-Hendrik Evertse, Kalman Gyory

CAMBRIDGE STUDIES IN ADVANCED MATHEMATICS 146
EditorialBoard
B. BOLLOBA´S, W. FULTON, A. KATOK, F. KIRWAN,
P. SARNAK, B. SIMON, B. TOTARO
UNITEQUATIONS IN
DIOPHANTINENUMBERTHEORY
Diophantinenumbertheoryisanactiveareathathasseentremendousgrowthoverthe
pastcentury,andinthistheoryunitequationsplayacentralrole.Thiscomprehensive
treatmentisthefirstvolumedevotedtotheseequations.Theauthorsgathertogetherall
themostimportantresultsandlookatmanydifferentaspects,includingeffectiveresults
onunitequationsovernumberfields,estimatesonthenumberofsolutions,analoguesfor
functionfields,andeffectiveresultsforunitequationsoverfinitelygenerateddomains.
Theyalsopresentavarietyofapplications.Introductorychaptersprovidethenecessary
backgroundinalgebraicnumbertheoryandfunctionfieldtheory,aswellasanaccount
oftherequiredtoolsfromDiophantineapproximationandtranscendencetheory.This
makesthebooksuitableforyoungresearchersaswellasforexpertswhoarelooking
foranup-to-dateoverviewofthefield.
Jan-Hendrik Evertse works at the Mathematical Institute of Leiden University. His
researchconcentratesonDiophantineapproximationandapplicationstoDiophantine
problems.Inthisareahehasobtainedsomeinfluentialresults,inparticularonestimates
forthenumbersofsolutionsofDiophantineequationsandinequalities.Hehaswritten
morethan75researchpapersandco-authoredonebookwithBasEdixhovenentitled
DiophantineApproximationandAbelianVarieties.
Ka´lma´nGyo˝ryisProfessorEmeritusattheUniversityofDebrecen,amemberofthe
HungarianAcademyofSciencesandawell-knownresearcherinDiophantinenumber
theory.Overhiscareerhehasobtainedseveralsignificantandpioneeringresults,among
othersonunitequations,decomposableformequations,andtheirvariousapplications.
Hisresultshavebeenpublishedinonebookand160researchpapers.Gyo˝ryisalsothe
founderandleaderofthenumbertheoryresearchgroupinDebrecen,whichconsistsof
hisformerstudentsandtheirstudents.
CAMBRIDGE STUDIES IN ADVANCED MATHEMATICS
EditorialBoard:
B.Bolloba´s,W.Fulton,A.Katok,F.Kirwan,P.Sarnak,B.Simon,B.Totaro
AllthetitleslistedbelowcanbeobtainedfromgoodbooksellersorfromCambridgeUniversityPress.
Foracompleteserieslistingvisitwww.cambridge.org/mathematics.
Alreadypublished
109 H.GeigesAnintroductiontocontacttopology
110 J.FarautAnalysisonLiegroups:Anintroduction
111 E.ParkComplextopologicalK-theory
112 D.W.StroockPartialdifferentialequationsforprobabilists
113 A.Kirillov,Jr AnintroductiontoLiegroupsandLiealgebras
114 F.Gesztesyetal.Solitonequationsandtheiralgebro-geometricsolutions,II
115 E.deFaria&W.deMeloMathematicaltoolsforone-dimensionaldynamics
116 D.ApplebaumLe´vyprocessesandstochasticcalculus(2ndEdition)
117 T.SzamuelyGaloisgroupsandfundamentalgroups
118 G.W.Anderson,A.Guionnet&O.ZeitouniAnintroductiontorandommatrices
119 C.Perez-Garcia&W.H.SchikhofLocallyconvexspacesovernon-Archimedeanvaluedfields
120 P.K.Friz&N.B.VictoirMultidimensionalstochasticprocessesasroughpaths
121 T.Ceccherini-Silberstein,F.Scarabotti&F.TolliRepresentationtheoryofthesymmetricgroups
122 S.Kalikow&R.McCutcheonAnoutlineofergodictheory
123 G.F.Lawler&V.LimicRandomwalk:Amodernintroduction
124 K.Lux&H.PahlingsRepresentationsofgroups
125 K.S.Kedlayap-adicdifferentialequations
126 R.Beals&R.WongSpecialfunctions
127 E.deFaria&W.deMeloMathematicalaspectsofquantumfieldtheory
128 A.TerrasZetafunctionsofgraphs
129 D.Goldfeld&J.HundleyAutomorphicrepresentationsandL-functionsforthegenerallineargroup,I
130 D.Goldfeld&J.HundleyAutomorphicrepresentationsandL-functionsforthegenerallineargroup,II
131 D.A.CravenThetheoryoffusionsystems
132 J.Va¨a¨na¨nenModelsandgames
133 G.Malle&D.TestermanLinearalgebraicgroupsandfinitegroupsofLietype
134 P.LiGeometricanalysis
135 F.MaggiSetsoffiniteperimeterandgeometricvariationalproblems
136 M.Brodmann&R.Y.SharpLocalcohomology(2ndEdition)
137 C.Muscalu&W.SchlagClassicalandmultilinearharmonicanalysis,I
138 C.Muscalu&W.SchlagClassicalandmultilinearharmonicanalysis,II
139 B.HelfferSpectraltheoryanditsapplications
140 R.Pemantle&M.C.WilsonAnalyticcombinatoricsinseveralvariables
141 B.Branner&N.FagellaQuasiconformalsurgeryinholomorphicdynamics
142 R.M.DudleyUniformcentrallimittheorems(2ndEdition)
143 T.LeinsterBasiccategorytheory
144 I.Arzhantsev,U.Derenthal,J.Hausen&A.LafaceCoxrings
145 M.VianaLecturesonLyapunovexponents
146 J.-H.Evertse&K.Gyo˝ryUnitequationsinDiophantinenumbertheory
147 A.PrasadRepresentationtheory
148 S.R.Garcia,J.Mashreghi&W.T.RossIntroductiontomodelspacesandtheiroperators
Unit Equations in
Diophantine Number Theory
JAN-HENDRIK EVERTSE
UniversiteitLeiden
KA´ LMA´ N GYO˝ RY
DebreceniEgyetem,Hungary
UniversityPrintingHouse,CambridgeCB28BS,UnitedKingdom
CambridgeUniversityPressispartoftheUniversityofCambridge.
ItfurtherstheUniversity’smissionbydisseminatingknowledgeinthepursuitof
education,learningandresearchatthehighestinternationallevelsofexcellence.
www.cambridge.org
Informationonthistitle:www.cambridge.org/9781107097605
©Jan-HendrikEvertseandKa´lma´nGyo˝ry2015
Thispublicationisincopyright.Subjecttostatutoryexception
andtotheprovisionsofrelevantcollectivelicensingagreements,
noreproductionofanypartmaytakeplacewithoutthewritten
permissionofCambridgeUniversityPress.
Firstpublished2015
PrintedintheUnitedKingdombyClays,StIvesplc
AcataloguerecordforthispublicationisavailablefromtheBritishLibrary
ISBN978-1-107-09760-5Hardback
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ofURLsforexternalorthird-partyinternetwebsitesreferredtointhispublication,
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accurateorappropriate.
Contents
Preface pageix
Summary xi
PART I PRELIMINARIES
1 Basicalgebraicnumbertheory 3
1.1 Characteristicpolynomial,trace,norm,discriminant 3
1.2 Idealtheoryforalgebraicnumberfields 5
1.3 Extensionofideals;normofideals 7
1.4 Discriminant,classnumber,unitgroupandregulator 9
1.5 Explicitestimates 11
1.6 Absolutevalues:generalities 12
1.7 Absolutevaluesandplacesonnumberfields 15
1.8 S-integers,S-unitsandS-norm 17
1.9 Heights 19
1.9.1 Heightsofalgebraicnumbers 19
1.9.2 v-adicnormsandheightsofvectors
andpolynomials 21
1.10 Effectivecomputationsinnumberfields 23
1.11 p-adicnumbers 26
2 Algebraicfunctionfields 30
2.1 Valuations 30
2.2 Heights 33
2.3 Derivativesandgenus 35
2.4 Effectivecomputations 37
3 ToolsfromDiophantineapproximationand
transcendencetheory 42
v
vi Contents
3.1 TheSubspaceTheoremandsomevariations 42
3.2 Effectiveestimatesforlinearformsinlogarithms 51
PART II UNIT EQUATIONS AND APPLICATIONS
4 Effectiveresultsforunitequationsintwounknownsover
numberfields 61
4.1 Effectiveboundsfortheheightsofthesolutions 62
4.1.1 Equationsinunitsofanumberfield 62
4.1.2 Equationswithunknownsfromafinitely
generatedmultiplicativegroup 64
4.2 Approximationbyelementsofafinitelygenerated
multiplicativegroup 67
4.3 Tools 68
4.3.1 Somegeometryofnumbers 68
4.3.2 EstimatesforunitsandS-units 72
4.4 Proofs 79
4.4.1 ProofsofTheorems4.1.1and4.1.2 79
4.4.2 ProofsofTheorems4.2.1and4.2.2 81
4.4.3 ProofsofTheorem4.1.3anditscorollaries 84
4.5 Alternativemethods,comparisonofthebounds 87
4.5.1 TheresultsofBombieri,BombieriandCohen,
andBugeaud 87
4.5.2 TheresultsofMurty,PastenandvonKa¨nel 88
4.6 Theabc-conjecture 89
4.7 Notes 93
4.7.1 Historicalremarksandsomerelatedresults 93
4.7.2 Somenotesonapplications 94
5 Algorithmicresolutionofunitequationsintwounknowns 96
5.1 ApplicationofBaker’stypeestimates 97
5.1.1 Infiniteplaces 100
5.1.2 Finiteplaces 102
5.2 Reductionofthebounds 103
5.2.1 Infiniteplaces 103
5.2.2 Finiteplaces 105
5.3 Enumerationofthe“small”solutions 111
5.4 Examples 119
5.5 Exceptionalunits 121
5.6 Supplement:LLLlatticebasisreduction 123
5.7 Notes 126
Contents vii
6 Unitequationsinseveralunknowns 128
6.1 Results 130
6.1.1 Asemi-effectiveresult 130
6.1.2 Upperboundsforthenumberofsolutions 131
6.1.3 Lowerbounds 134
6.2 ProofsofTheorem6.1.1andCorollary6.1.2 136
6.3 AsketchoftheproofofTheorem6.1.3 140
6.3.1 Areduction 140
6.3.2 Notation 142
6.3.3 Coveringresults 142
6.3.4 Thelargesolutions 144
6.3.5 Thesmallsolutions,andconclusionoftheproof 147
6.4 ProofofTheorem6.1.4 148
6.5 ProofofTheorem6.1.6 158
6.6 ProofsofTheorems6.1.7and6.1.8 161
6.7 Notes 165
7 Analoguesoverfunctionfields 173
7.1 Mason’sinequality 174
7.2 Proofs 176
7.3 Effectiveresultsinthemoreunknownscase 178
7.4 Resultsonthenumberofsolutions 182
7.5 ProofofTheorem7.4.1 183
7.5.1 Extensiontothek-closureof(cid:2) 183
7.5.2 Somealgebraicgeometry 185
7.5.3 ProofofTheorem7.5.1 188
7.6 Resultsinpositivecharacteristic 192
8 Effectiveresultsforunitequationsintwounknownsover
finitelygenerateddomains 197
8.1 Statementsoftheresults 198
8.2 Effectivelinearalgebraoverpolynomialrings 201
8.3 Areduction 204
8.4 BoundingthedegreeinProposition8.3.7 212
8.5 Specializations 215
8.6 BoundingtheheightinProposition8.3.7 222
8.7 ProofofTheorem8.1.3 225
8.8 Notes 230
9 Decomposableformequations 231
9.1 Afinitenesscriterionfordecomposableformequations 233
viii Contents
9.2 Reductionofunitequationstodecomposable
formequations 236
9.3 Reductionofdecomposableformequationsto
unitequations 237
9.3.1 Proofoftheequivalence(ii)⇐⇒(iii)
inTheorem9.1.1 238
9.3.2 Proofoftheimplication(i)⇒(iii)in
Theorem9.1.1 238
9.3.3 Proofoftheimplication(iii)⇒(i)in
Theorem9.1.1 240
9.4 Finitenessofthenumberoffamiliesofsolutions 244
9.5 Upperboundsforthenumberofsolutions 249
9.5.1 GaloissymmetricS-unitvectors 251
9.5.2 Consequencesfordecomposableformequations
andS-unitequations 253
9.6 Effectiveresults 257
9.6.1 Thueequations 258
9.6.2 Decomposableformequationsinanarbitrary
numberofunknowns 263
9.7 Notes 272
10 Furtherapplications 284
10.1 Primefactorsofsumsofintegers 284
10.2 Additiveunitrepresentationsinfinitelygenerated
integraldomains 287
10.3 Orbitsofpolynomialandrationalmaps 291
10.4 Polynomialsdividingmanyk-nomials 298
10.5 Irreduciblepolynomialsandarithmeticgraphs 301
10.6 Discriminantequationsandpowerintegralbasesin
numberfields 305
10.7 Binaryformsofgivendiscriminant 310
10.8 Resultantequationsformonicpolynomials 315
10.9 Resultantinequalitiesandequationsforbinaryforms 317
10.10 Lang’sConjecturefortori 321
10.11 Linearrecurrencesequencesand
exponential-polynomialequations 326
10.12 Algebraicindependenceresults 330
References 337
Glossaryoffrequentlyusednotation 358
Index 361
Preface
Diophantinenumbertheory(thestudyofDiophantineequations,Diophantine
inequalitiesandtheirapplications)isaveryactiveareainnumbertheorywitha
longhistory.Thisbookisaboutunitequations,aclassofDiophantineequations
of central importance in Diophantine number theory, and their applications.
Unitequationsareequationsoftheform
a x +···+a x =1
1 1 n n
to be solved in elements x ,...,x from a finitely generated multiplicative
1 n
group (cid:2), contained in a field K, where a ,...,a are non-zero elements of
1 n
K. Such equations were studied originally in the cases where the number
of unknowns n=2, K is a number field and (cid:2) is the group of units of the
ring of integers of K, or more generally, where (cid:2) is the group of S-units
in K. Unit equations have a great variety of applications, among others to
other classes of Diophantine equations, to algebraic number theory and to
Diophantinegeometry.
Certainresultsconcerningunitequationsandtheirapplicationscoveredin
ourbookwerealreadypresented,mostlyinspecialorweakerform,inthebooks
ofLang(1962,1978,1983),Gyo˝ry(1980b),Sprindzˇuk(1982,1993),Evertse
(1983),Mason(1984),ShoreyandTijdeman(1986),deWeger(1989),Schmidt
(1991),Smart(1998),BombieriandGubler(2006),BakerandWu¨stholz(2007)
and Zannier (2009), and in the survey papers of Evertse, Gyo˝ry, Stewart and
Tijdeman (1988b), Gyo˝ry (1992a, 1996, 2002a, 2010) and Be´rczes, Evertse
andGyo˝ry(2007b).
In1988,wewrote,togetherwithStewartandTijdeman,thesurveyEvertse,
Gyo˝ry,StewartandTijdeman(1988b)onunitequationsandtheirapplications
giving the state of the art of the subject at that time. Since then, the theory
of unit equations has been greatly expanded. In the present book we have
ix