CONGRUENCES AND LEHMER’S PROBLEM1.pdf

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1、July31,200814:36WSPC/203-IJNT00154InternationalJournalofNumberTheoryVol.4,No.4(2008)587–596
cWorldScientificPublishingCompanyCONGRUENCESANDLEHMER’SPROBLEMNICOLAECIPRIANBONCIOCATInstituteofMathematicsoftheRomanianAcademyP.O.Box1-764,Bucharest014700,RomaniaNicolae

2、.Bonciocat@imar.roReceived9October2006Accepted19April2007WeobtainexplicitlowerboundsfortheMahlermeasurefornonreciprocalpolynomialswithintegercoefficientssatisfyingcertaincongruences.Keywords:Mahlermeasure;nonreciprocalpolynomial.MathematicsSubjclassClassification2

3、000:11R09,11C08,11Y401.IntroductionTheMahlermeasureofapolynomial
nnf(z)=azi=a(z−α)inii=0i=1isdefinedbynM(f)=

4、an

5、max{1,

6、αi

7、}.i=1Asweknow,M(f)=M(f∗),wheref∗(z)=znf(1/z)isthereciprocaloff,andapolynomialissaidtobereciprocaliff(z)=±f∗(z).ByaclassicalresultofKro-nec

8、ker,M(f)=1ifandonlyiff(z)isaproductofcyclotomicpolynomialsandapoweroftheindeterminatez.Along-standingopenproblemduetoLehmer[5]askswhetherthereisagapbetween1andthenextlargestalgebraicintegerwithrespecttotheMahlermeasure,orequivalently,ifforany>0thereexistsapoly

9、nomialfwithintegercoefficientssuchthat1

10、andZassen-hausin[8].Theyconjecturedthatthereexistsapositiveconstantcsuchthatevery587July31,200814:36WSPC/203-IJNT00154588N.C.Bonciocatmonic,irreduciblepolynomialofdegreenhasarootαsatisfying

11、α

12、>1+c/n,andonemayeasilycheckthatsolvingLehmer’sproblemresolvesthisprob

13、lemaswell.Smyth[9]answeredLehmer’squestionforthecaseofnonreciprocalpolynomialsbyprovingthatM(f)≥M(z3−z−1)=1.324717...foranynonreciprocalpolynomialfwithf(0)=0.Schinzel[6]provedthatiffisamonic,integerpolynomialofdegreedsatisfyingf(0)=±1andf(±1)=0,andallrootsoff

14、arereal,thenM(f)≥γd/2,√whereγ=(1+5)/2,thegoldenratio.AnotherelegantresultwasprovedbyBor-niwein,HareandMossinghoff[3]fornonreciprocalpolynomialsf(z)=i=0aizsat-isfyingf≡±f∗mod