ebook img

Andrzej Schinzel, Selecta (Heritage of European Mathematics) PDF

1419 Pages·2007·9.42 MB·English
Save to my drive
Quick download
Download
Most books are stored in the elastic cloud where traffic is expensive. For this reason, we have a limit on daily download.

Preview Andrzej Schinzel, Selecta (Heritage of European Mathematics)

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) The Swiss National Library lists this publication in The Swiss Book,the Swiss national bibliography, and the detailed bibliographic data are available on the Internet at http://www.helveticat.ch. This work is subject to copyright.All rights are reserved,whether the whole or part of the material is concerned,specifically the rights of translation,reprinting,re-use of illustrations,recitation,broadca- sting,reproduction on microfilms or in other ways,and storage in data banks.For any kind of use per- mission of the copyright owner must be obtained. © 2007 European Mathematical Society Contact address: European Mathematical Society Publishing House Seminar for Applied Mathematics ETH-Zentrum FLI C4 CH-8092 Zürich Switzerland Phone:+41 (0)44 632 34 36 Email:[email protected] Homepage:www.ems-ph.org Printed in Germany 9 8 7 6 5 4 3 2 1 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

Description:
Andrzej Schinzel, born in 1937, is a leading number theorist whose work has had a lasting impact on modern mathematics. He is the author of over 200 research articles in various branches of arithmetics, including elementary, analytic, and algebraic number theory. He has also been, for nearly 40 year
See more

The list of books you might like

Most books are stored in the elastic cloud where traffic is expensive. For this reason, we have a limit on daily download.