Selecta. 2 Vols: Diophantine problems and polynomials

872 Pages·2007·5 MB·English

by  Andrzej Schnizel

schinzel_vol_1_titelei 5.3.2007 18:32 Uhr Seite 1
H e r i t a g e o f E u r o p e a n M a t h e m a t i c s
Advisory Board
Michèle Audin, Strasbourg
Ciro Ciliberto, Roma
Ildar A. Ibragimov, St. Petersburg
Wladyslaw Narkiewicz, Wroclaw
Peter M. Neumann, Oxford
Samuel J. Patterson, Göttingen
schinzel_vol_1_titelei 5.3.2007 18:33 Uhr Seite 2
Andrzej Schinzel in Cambridge (England), 1961
schinzel_vol_1_titelei 5.3.2007 18:33 Uhr Seite 3
Andrzej Schinzel
Selecta
Volume I
Diophantine Problems and Polynomials
Edited by
Henryk Iwaniec
Wladyslaw Narkiewicz
Jerzy Urbanowicz
schinzel_vol_1_titelei 5.3.2007 18:33 Uhr Seite 4
Author:
Andrzej Schinzel
Institute of Mathematics
Polish Academy of Sciences
ul.Śniadeckich 8,skr.poczt.21
00-956 Warszawa 10
Poland
Editors:
Henryk Iwaniec Władysław Narkiewicz Jerzy Urbanowicz
Department of Mathematics Institute of Mathematics Institute of Mathematics
Rutgers University University of Wrocław Polish Academy of Sciences
New Brunswick,NJ 08903 pl.Grunwaldzki 2/4 ul.Śniadeckich 8,skr.poczt.21
U.S.A. 50-384 Wrocław 00-956 Warszawa 10
[email protected] Poland Poland
[email protected] [email protected]
2000 Mathematics Subject Classification:11,12
ISBN 978-3-03719-038-8 (Set Vol I & Vol II)
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Preface
AndrzejBobolaMariaSchinzel,bornonApril5,1937atSandomierz(Poland),iswell
known for his original results in various areas of number theory appearing in over 200
research papers, of which the first thirty were published while he was an undergraduate
attheWarsawUniversity.WorkingundertheguidanceofWacławSierpin´ski,hebecame
interested in elementary number theory, and the subjects of his early papers range from
propertiesofarithmeticalfunctions,likeEuler’sϕ-functionorthenumberofdivisors,to
Diophantineequations.PaulErdo˝s,abigchampionofelementarynumbertheory,wrotein
hislettertoSierpin´skiofOctober23,1960—whenAndrzejwasstudyingattheUniversity
ofCambridgeunderthesupervisionofHaroldDavenport—“Schinzel’scompletionofmy
proof is much simpler than anything I had in mind”. Many mathematicians cooperating
withAndrzejSchinzelcouldrepeatthewordsofErdo˝s.
SincecompletinghisstudiesatWarsawUniversityin1958,AndrzejSchinzelhasbeen
employedbytheInstituteofMathematicsofthePolishAcademyofSciences,wherehe
obtainedhisPh.D.in1960.OnhisreturnfromaRockefellerFoundationFellowshipatthe
UniversityofCambridgeandtheUniversityofUppsala(wherehestudiedunderTrygve
Nagell) he completed his habilitation in 1962. In 1967 he was promoted to Associate
Professor,andin1974toFullProfessor.In1979hewaselectedtoCorrespondingMember
ofthePolishAcademyofSciencesandin1994toFullMember.
AndrzejSchinzel’sveryfirstpaper(1)—publishedattheageof17—isapostscriptto
a result of H.-E. Richert who proved a general theorem about partitions of integers into
distinctsummandsfromagivensetwhichimpliedinparticularthateveryinteger>33isa
sumofdistincttriangularnumbers.Schinzelobservedthateveryinteger>51isasumofat
mostfourdistincttriangularnumbers.ThefavoritesubjectoftheearlyresearchofSchinzel,
Euler’s totient function, is considered here in five papers; the earliest one, published in
1954, is not included(2). In 1958 a joint work J1 with Sierpin´ski appeared, analyzing
various consequences of the conjecture stating that if f ,...,f ∈ Z[x] are irreducible
1 s
polynomialshavingpositiveleadingcoefficientsandthereisnonaturalnumber> 1that
isadivisorofeachofthenumbersf (n)···f (n)fornbeinganintegerthenforinfinitely
1 s
manynaturalnthevaluesf (n),...,f (n)areprimes.Thiscelebratedconjecture—with
1 s
manyunexpectedconsequences—iscalled“Schinzel’sHypothesisH”.
Schinzel’sdoctoralthesisB1dealtwiththeperiodofaclassofcontinuedfractionsand
wasrelated(seeB2)toaquestionconcerningpseudo-ellipticintegrals,consideredalready
(1) Surladécompositiondesnombresnaturelsensommesdenombrestriangulairesdistincts,Bull.
Acad.Polon.Sci.Cl.III2(1954),409–410.
(2) Surquelquespropriétésdesfonctionsϕ(n)etσ(n),Bull.Acad.Polon.Sci.Cl.III2(1954),
463–466(withW.Sierpin´ski).
vi Preface
by N. H. Abel in the very first volume of Crelle’s Journal. In his habilitation thesis—
consistingoffourpapersI1,I2,I3andapapernotincluded(3)—Schinzelgeneralizeda
classicaltheoremofZsigmondyof1892(oftencalledtheBirkhoff–Vandivertheorem)on
primitivedivisors.
ThecentralthemeofSchinzel’sworkisarithmeticalandalgebraicpropertiesofpoly-
nomialsinoneorseveralvariables,inparticularquestionsofirreducibilityandzerosof
polynomials. To this topic he devoted about one-third of his papers and two books(4).
In the books Schinzel presented several classical results and included many extensions,
improvementsandgeneralizationsofhisown.
Undoubtedly Schinzel and his beloved journal Acta Arithmetica influenced many
mathematicians, stimulating their thinking and mathematical careers. Andrzej Schinzel
has since 1969 been the editor of this first international journal devoted exclusively to
number theory, being a successor of his teacherW. Sierpin´ski.Among the other editors
ofActaArithmeticaduringtheseyearswere/areJ.W.S.Cassels,H.Davenport,P.Erdo˝s,
V.Jarník,J.Kaczorowski,Yu.V.Linnik,L.J.Mordell,W.M.Schmidt,V.G.Sprindzhuk,
R.TijdemanandP.Turán.Thesepeopleandtheotheroutstandingmathematiciansfrom
theadvisoryboardofActaArithmeticahavedeterminedthelineofthejournal.
AndrzejSchinzel’sworkhasbeeninfluentialinthedevelopmentofmanyareasofmath-
ematics,andhis70thbirthdaygivesusanopportunitytohonorhisaccomplishmentsby
puttingtogetherhismostimportantpapers.ThisselectionofSchinzel’spapers—published
duringmorethanfivedecades—isdividedintotwovolumescontaining100articles.We
haveaskedsomeoutstandingmathematiciansforcommentariestotheselectedpapers.Also
includedisalistofunsolvedproblemsandunprovedconjecturesproposedbySchinzelin
theyears1956–2006,arrangedchronologically.Thefirstvolumecoverssixthemes:
A. Diophantineequationsandintegralforms(withcommentariesbyRobertTijdeman)
B. Continuedfractions(withcommentariesbyEugèneDubois)
C. Algebraicnumbertheory(withcommentariesbyDavidW.BoydandDonaldJ.Lewis)
D. Polynomialsinonevariable(withcommentariesbyMichaelFilaseta)
E. Polynomialsinseveralvariables(withcommentariesbyUmbertoZannier)
F. Hilbert’sIrreducibilityTheorem(withcommentariesbyUmbertoZannier)
Thesecondvolumecontainspaperscoveringseventhemes:
G. Arithmeticfunctions(withcommentariesbyKevinFord)
H. Divisibilityandcongruences(withcommentariesbyHendrikW.Lenstra,Jr.)
I. Primitivedivisors(withcommentariesbyCameronL.Stewart)
J. Primenumbers(withcommentariesbyJerzyKaczorowski)
K. Analyticnumbertheory(withcommentariesbyJerzyKaczorowski)
L. Geometryofnumbers(withcommentariesbyWolfgangM.Schmidt)
M. Otherpapers(withcommentariesbyStanisławKwapien´ andEndreSzemerédi)
(3) The intrinsic divisors of Lehmer numbers in the case of negative discriminant,Ark. Mat. 4
(1962),413–416.
(4) Selected Topics on Polynomials, XXII+250 pp., University of Michigan Press, Ann Arbor
1982, and Polynomials with Special Regard to Reducibility, X+558 pp., Encyclopaedia of
MathematicsanditsApplications77,CambridgeUniv.Press,Cambridge2000.
Preface vii
Manypeoplehelpedwiththeeditingofthevolumes.Firstofall,wegratefullythank
the authors of the commentaries we had the pleasure to work with. Second, our special
thanksgotoStanisławJaneczko,DirectoroftheInstituteofMathematics,PolishAcademy
ofSciences,forhissupport,andtoManfredKarbe,PublishingDirectoroftheEuropean
MathematicalSocietyPublishingHouse,forhisinvaluableassistanceduringtheworkon
theSelecta.Third,wewishtothankJerzyBrowkinforreadingthepapersandsomecor-
rections, and Jan K. Kowalski for retyping the papers and offering valuable suggestions
for improving the presentation of the material. Finally, we wish to express our grati-
tude to the staff of the European Mathematical Society Publishing House, especially to
IreneZimmermann,fortheverypleasantcooperation.
Wehavedecidedtounifysomenotations,usingthe“blackboardbold”typeformost
commonsets;soC,R,Q,Q andF alwaysstandforthecomplex,real,rational,p-adic
p p
andfinitefieldsrespectively;Z,Z fortheringsofintegersandp-adicintegers;andN,N
p 0
forthesetsofpositiveandnonnegativeintegers.Thegreatestcommondivisorofintegers
a ,a ,...,a is denoted by (a ,a ,...,a ). If (a ,a ,...,a ) = 1, the integers are
1 2 n 1 2 n 1 2 n
calledrelativelyprime,andif(a ,a )=1forany1(cid:2)i (cid:3)=j (cid:2)n,theintegersarecalled
i j
coprime.Asusual,forx ∈Rset[x]=max{n∈Z:n(cid:2)x},(cid:4)x(cid:5)=min{n∈Z:x (cid:2)n}
and(cid:6)x(cid:6)=min{x−[x],(cid:4)x(cid:5)−x}.Wedenoteby{x}thefractionalpartofx.Lineswhere
minorcorrectionsoftheoriginaltexthavebeenmadearemarked“c”intheleftmargin.
March2007 HenrykIwaniec
WładysławNarkiewicz
JerzyUrbanowicz

Contents
Volume 1
A. Diophantineequationsandintegralforms 1
CommentaryonA:Diophantineequationsandintegralforms
byR.Tijdeman . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
A1 SurlesnombresdeMersennequisonttriangulaires
avecGeorgesBrowkin . . . . . . . . . . . . . . . . . . . . . . . . . . 11
A2 Surquelquespropriétésdesnombres3/net4/n,oùnestunnombreimpair 13
A3 Surl’existenced’uncerclepassantparunnombredonnédepointsaux
coordonnéesentières . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
A4 Surlessommesdetroiscarrés . . . . . . . . . . . . . . . . . . . . . . . . 18
(cid:2)
A5 OntheDiophantineequation n A xϑk =0 . . . . . . . . . . . . . . . 22
k=1 k k
A6 Polynomialsofcertainspecialtypes
withH.DavenportandD.J.Lewis . . . . . . . . . . . . . . . . . . . . 27
A7 AnimprovementofRunge’stheoremonDiophantineequations . . . . . . 36
A8 Ontheequationym =P(x)
withR.Tijdeman . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
A9 Zetafunctionsandtheequivalenceofintegralforms
withR.Perlis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
A10 QuadraticDiophantineequationswithparameters
withD.J.Lewis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
A11 Selmer’sconjectureandfamiliesofellipticcurves
withJ.W.S.Cassels . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
A12 Familiesofcurveshavingeachanintegerpoint . . . . . . . . . . . . . . . 67
A13 Hasse’sprincipleforsystemsofternaryquadraticformsandforone
biquadraticform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
A14 OnRunge’stheoremaboutDiophantineequations
withA.Grytczuk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
A15 Onsumsofthreeunitfractionswithpolynomialdenominators . . . . . . . 116
A16 Onequationsy2 =xn+kinafinitefield
withM.Skałba . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
x Contents
B. Continuedfractions 127
CommentaryonB:Continuedfractions
byEugèneDubois . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
B1 Onsomeproblemsofthearithmeticaltheoryofcontinuedfractions . . . . 131
B2 OnsomeproblemsofthearithmeticaltheoryofcontinuedfractionsII . . . 149
B3 OntwoconjecturesofP.ChowlaandS.Chowlaconcerningcontinued
fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
C. Algebraicnumbertheory 167
CommentaryonC:Algebraicnumbers
byDavidW.BoydandD.J.Lewis . . . . . . . . . . . . . . . . . . . . 169
C1 ArefinementoftwotheoremsofKronecker
withH.Zassenhaus . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
C2 OnatheoremofBauerandsomeofitsapplications . . . . . . . . . . . . 179
C3 AnextensionofthetheoremofBauerandpolynomialsofcertainspecialtypes
withD.J.LewisandH.Zassenhaus . . . . . . . . . . . . . . . . . . . 190
C4 Onsumsofrootsofunity.(SolutionoftwoproblemsofR.M.Robinson) . 197
C5 OnatheoremofBauerandsomeofitsapplicationsII . . . . . . . . . . . 210
C6 Ontheproductoftheconjugatesoutsidetheunitcircleofanalgebraicnumber 221
C7 Onlineardependenceofroots . . . . . . . . . . . . . . . . . . . . . . . . 238
C8 OnSylow2-subgroupsofK O forquadraticnumberfieldsF
2 F
withJ.Browkin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253
C9 Aclassofalgebraicnumbers . . . . . . . . . . . . . . . . . . . . . . . . 264
C10 OnvaluesoftheMahlermeasureinaquadraticfield(solutionofaproblem
ofDixonandDubickas) . . . . . . . . . . . . . . . . . . . . . . . . . 272
D. Polynomialsinonevariable 281
CommentaryonD:Polynomialsinonevariable
byMichaelFilaseta . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283
D1 Solutiond’unproblèmedeK.Zarankiewiczsurlessuitesdepuissances
consécutivesdenombresirrationnels . . . . . . . . . . . . . . . . . . 295
D2 Onthereducibilityofpolynomialsandinparticularoftrinomials . . . . . 301
D3 Reducibilityofpolynomialsandcoveringsystemsofcongruences . . . . . 333
D4 ReducibilityoflacunarypolynomialsI . . . . . . . . . . . . . . . . . . . 344
D5 ReducibilityoflacunarypolynomialsII . . . . . . . . . . . . . . . . . . . 381
D6 Anoteonthepaper“ReducibilityoflacunarypolynomialsI”
withJ.Wójcik . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403
D7 ReducibilityoflacunarypolynomialsIII . . . . . . . . . . . . . . . . . . 409
D8 ReducibilityoflacunarypolynomialsIV . . . . . . . . . . . . . . . . . . 447
D9 Onthenumberoftermsofapowerofapolynomial . . . . . . . . . . . . 450
D10 Onreducibletrinomials . . . . . . . . . . . . . . . . . . . . . . . . . . . 466