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ONTHEREMAKHEIGHT,THEMAHLERMEASURE,ANDCONJUGATESETSOFALGEBRAICNUMBERSLYINGONTWOCIRCLESA.DUBICKASandC.J.SMYTHWede neanewheightfunctionR( ),theRemakheightofanalgebraicnumber .WegivesharpupperandlowerboundsforR( )intermsoftheclassicalMahlermeasureM( ):Studyofwhenoneoftheseboundsisexactleadsustoconsiderationofconjugatesetsofalgebraicnumbersofnorm 1lyingontwocirclescenteredat0.Wegiveacompletecharacterisationofsuchconjugatesets.Theyturnouttobeoftwotypes:onerelatedtocertaincubicalgebraicnumbers,andtheotherrelatedtoanon-integergeneralisationofSalemnumberswhichwecallextendedSalemnumbers.1991MathematicsSubjectClassi cation:11R06:1.Introduction.Let beanalgebraicnumber,ofdegreed2,withminimalpolynomiala0zd+:::+ad2Z[z]overtherationals,conjugates 1; 2;:::; d(with oneofthese)labelledsothatj 1jj 2j:::j dj.In1952RobertRemak[Re]gaveanewupperboundforthemodulusofaVandermondedeterminant.Whenappliedtothediscriminant =a2d 20 ij( i j)2,hisboundispj j6dd=2ja0jd 1j 1jd 1j 2jd 2:::j d 1j:AsweshallseefromTheorem1below,thisboundisatleastasstrongastheclassicalboundpj j6dd=2M( )d 1(1)whichcomesstraightfromHadamard sinequality([H],[HLP],p.34);sometimesRemak sboundissigni cantlystronger.HereM( )istheMahlermeasure(height)M( )=ja0jd i=1max(1;j ij):(2)SeeforinstanceEverestandWard[EW]foranintroductiontoMahlermeasure.Accordingly,wede netheRemakheightR( )byR( )=ja0jj 1jj 2jd 2d 1j 3jd 3d 1:::j d 1j1d 1;(3)1sothatRemak’sboundcanbewrittenpj j6dd=2R( )d 1;(4)resembling(1).InthispaperweobtainsharpupperandlowerboundsforR( )intermsofM( ),anddescribeinturnthose forwhicheachoftheseboundsisinfactanequality.Thisleadsustothestudyofalgebraicnumberslyingwiththeirconjugatesontwocircles(Theorem2).Here,andthroughoutthepaper,allcirclesareassumedtobecenteredat0.We rstneedsomede nitions.RecallthataSalemnumberisanalgebraicinteger 1ofdegree4conjugateto 1whoseotherconjugatesalllieonjzj=1.De neanextendedSalemnumbertobeanalgebraicnumber 1,ofdegreeatleast4,conjugateto 1,whoseotherconjugatesalllieonjzj=1.SotheextendedSalemnumberswhicharealgebraicintegersaretheSalemnumbers.yAnalgebraicnumber isreciprocalif 1isaconjugateof .Saythatanalgebraicnumber isaunit-circularif lieswithitsconjugatesonjzj=1.So,byKronecker’sTheorem([EW],p.27),theunit-circularalgebraicintegersaretherootsofunity.Wenowstateour rstmainresult.Theorem1.Let beanalgebraicnumberofdegreed2,andminimalpolynomiala0zd+:::+adovertherationals.ThenM( )dd 1min(ja0j;jadj)(max(ja0j;jadj))1=(d 1)!1=26R( )6M( ):(5)The rstinequalityisanequalitypreciselywheneither(i)ja0j=jadjand lieswithitsconjugatesontwocircles(butnotonjustone)or(ii) lieswithitsconjugatesononecircle.Thesecondinequalityisanequalitypreciselywheneither(iii)d=2andj 1j1j 2jor(iv)d4and iseitherunit-circularoris anextendedSalemnumber.Notes.1.SinceM( )max(ja0j;jadj),itfollowsfrom(5)thatalsoR( )pM( )min(ja0j;jadj)(6)yIfSalemnumberswererenamedSalemintegers,thenextendedSalemnumberscouldsimplybecalledSalemnumbers!2andsocertainlyR( )pja0adj1:Thislastinequalitywasprovedealierin[D2].If isnotaunit,thenR( )p2,withR( )=p2for d= 2or 1=2.2.From(5)and(6),R( )=1i M( )=1.SoR( )=1i isarootofunity.3.Inequality(5)infactholdsforanyP2C[z]witha0ad6=0andM( );R( )replacedbytheirpolynomialversionsM(P);R(P)de nedasin(2),(3)respectively,withthe ibeingthezerosofP.IfwerestrictTheorem1to beingaunit,weimmediatelyget:Theorem10.Let beaunitofdegreed2.ThenM( )d2(d 1)6R( )6M( ):(50)The rstinequalityisanequalitywheneither lieswithitsconjugatesontwocircles,or iscyclotomic.Thesecondinequalityisanequalitywheneither(i)d=2or(ii)d4and iseithercyclotomicor aSalemnumber.2.Conjugatesetsofalgebraicnumbersoncircles:results.Inthissectionwedescribethosealgebraicnumbers whichsatisfyeitherconditions(i)or(ii)ofTheorem1.Todothis,itisconvenientatthispointtomaketwomorenewde nitions.De neaunit-normtobeanalgebraicnumber withja0j=jadj.Sotheunit-normalgebraicintegersaretheunits.NotethatextendedSalemnumbersareunit-norms.Let beanextendedSalemnumberorareciprocalquadratic,ofdegree2s,withconjugateset 1; 12;:::; 1s .Wede neanalgebraicnumbertobeaSalemhalf-normif= 1 22::: ssforsomesuch andsomei= 1(i=1;:::;s).All2ssuchnumbersaredistinct(seeCorollary4).Notethatallconjugatesofsuchanareoftheform0= 1 22::: ssforsome i= 1(i=1;:::;s),andsolieonthetwocirclesjzj= andjzj= 1:Infact,asisaunit-norm(byLemma4),halfofitsconjugatesmustlieononecircle,andhalfontheother.Clearly,deg62s:Itneednotequal2showever;thecelebratedLehmerdegree2s=10example([EW],p.13)givesrisetotwonon-conjugateSalemhalf-norms,havingminimalpolynomialsP(z)andz16P(1=z),whereP(z)=z16+2z15+z14 2z13 4z12 2z11+3z10+5z9+3z8+z7 z5 z4 z3+z2+z+1:Wecannowstateoursecondmainresult,elucidatingcondition(i)ofTheorem1.Theorem2.Supposethataunit-norm ofdegreedlies,withallitsconjugates,ontwocirclesjzj=randjzj=R,butnotjustonone,with(withoutlossofgenerality)atmosthalfoftheconjugatesonjzj=r.Thenoneofthefollowingholds:3(a)disamultipleof3,R=r 1=2with havingd3conjugatesonjzj=rand2d3onjzj=r 1=2.Assuming(withoutlossofgenerality)thatj j=rwehave,furthermore,thatforsomepositiveinte