On rational approximation of algebraic functions

arXiv:math/0409353v2[math.CA]17Jun2005ONRATIONALAPPROXIMATIONOFALGEBRAICFUNCTIONSJULIUSBORCEA∗,RIKARDBØGVAD,ANDBORISSHAPIROAbstract.Weconstructanewschemeofapproximationofanymultivaluedalgebraicfunctionf(z)byasequence{rn(z)}n∈Nofrationalfunctions.Thelattersequenceisgeneratedbyarecurrencerelationwhichiscompletelyde-terminedbythealgebraicequationsatisfiedbyf(z).ComparedtotheusualPad´eapproximationourschemehasanumberofadvantages,suchassimplecomputationalproceduresthatallowustoprovenaturalanalogsofthePad´eConjectureandNuttall’sConjectureforthesequence{rn(z)}n∈Ninthecom-plementCP1\Df,whereDfistheunionofafinitenumberofsegmentsofrealalgebraiccurvesandfinitelymanyisolatedpoints.Inparticular,ourconstruc-tionmakesitpossibletocontrolthebehaviorofspuriouspolesandtodescribetheasymptoticratiodistributionofthefamily{rn(z)}n∈N.Asanapplicationwesettletheso-called3-conjectureofEgeciogluetaldealingwitha4-termrecursionrelatedtoapolynomialRiemannHypothesis.1.IntroductionandmainresultsRationalapproximantsofanalyticfunctionsandtheasymptoticdistributionoftheirzerosandpolesareofcentralinterestinmanyareasofmathematicsandphysics.FortheclassofalgebraicfunctionsthesequestionshaveattractedspecialattentionduetotheirimportantapplicationsrangingfromtheconvergencetheoryofPad´eapproximants[30]–[35]andthetheoryofgeneralorthogonalpolynomials[16,26,36,38]tostatisticalmechanics[28,29],complexSturm-Liouvilleproblems[6],inversescatteringtheoryandquantumfieldtheory[2].Themainpurposeofthispaperistogiveasimpleanddirectconstructionoffamiliesofrationalfunctionsconvergingtoacertainbranchofanarbitrary(multi-valued)algebraicfunctionf(z).WhiletheusualPad´eapproximationrequirestheknowledgeoftheTaylorexpansionoff(z)at∞,ourschemeisbasedonlyonthealgebraicequationsatisfiedbyf(z)andhasthereforeanessentiallydifferentrangeofapplications.Indeed,letP(y,z)=kXi=0Pk−i(z)yk−i(1.1)denotetheirreduciblepolynomialin(y,z)definingf(z),thatis,P(f(z),z)=0.NotethatP(y,z)isuniquelydefineduptoascalarfactor.Letusrewrite(1.1)as−yk=kXi=1Pk−i(z)Pk(z)yk−i(1.2)andconsidertheassociatedrecursionoflengthk+1withrationalcoefficients−qn(z)=kXi=1Pk−i(z)Pk(z)qn−i(z).(1.3)2000MathematicsSubjectClassification.Primary30E10;Secondary41A20,42C05,82B05.Keywordsandphrases.Finiterecursions,asymptoticratiodistribution,Pad´eapproximation.∗Correspondingauthor.12J.BORCEA,R.BØGVAD,ANDB.SHAPIROChoosinganyinitialk-tupleofrationalfunctionsIN={q0(z),...,qk−1(z)}onecangenerateafamily{qn(z)}n∈Nofrationalfunctionssatisfying(1.3)foralln≥kandcoincidingwiththeentriesofINfor0≤n≤k−1.Themainobjectofstudyofthispaperisthefamily{rn(z)}n∈N,wherern(z)=qn(z)qn−1(z).Inordertoformulateourresultsweneedseveraladditionaldefinitions.Considerfirstausualrecurrencerelationoflengthk+1withconstantcoefficients−un=α1un−1+α2un−2+...+αkun−k,(1.4)whereαk6=0.Definition1.Theasymptoticsymbolequationofrecurrence(1.4)isgivenbytk+α1tk−1+α2tk−2+...+αk=0.(1.5)Theleft-handsideoftheaboveequationiscalledthecharacteristicpolynomialofrecurrence(1.4).Denotetherootsof(1.5)byτ1,...,τkandcallthemthespectralnumbersoftherecurrence.Definition2.Recursion(1.4)anditscharacteristicpolynomialaresaidtobeofdominanttypeordominantforshortifthereexistsaunique(simple)spectralnumberτmaxofthisrecurrencerelationsatisfying|τmax|=max1≤i≤k|τi|.Other-wise(1.4)and(1.5)aresaidtobeofnondominanttypeorjustnondominant.Thenumberτmaxwillbereferredtoasthedominantspectralnumberof(1.4)orthedominantrootof(1.5).Thefollowingtheoremmaybefoundin[37,Ch.4].Theorem1.Letk∈Nandconsiderak-tuple(α1,...,αk)ofcomplexnumberswithαk6=0.Foranyfunctionu:Z≥0→Cthefollowingconditionsareequivalent:(i)Pn≥0untn=Q1(t)Q2(t),whereQ2(t)=1+α1t+α2t2+...+αktkandQ1(t)isapolynomialintwhosedegreeissmallerthank.(ii)Foralln≥kthenumbersunsatisfythe(k+1)-termrecurrencerelationgivenby(1.4).(iii)Foralln≥0onehasun=rXi=1pi(n)τni,(1.6)whereτ1,...,τrarethedistinctspectralnumbersof(1.4)withmultiplicitiesm1,...,mr,respectively,andpi(n)isapolynomialinthevariablenofdegreeatmostmi−1for1≤i≤r.NotethatbyDefinition2thedominantspectralnumberτmaxofanydominantrecurrencerelationhasmultiplicityone.Definition3.Aninitialk-tupleofcomplexnumbers{u0,u1,...,uk−1}iscalledfastgrowingwithrespecttoagivendominantrecurrenceoftheform(1.4)ifthecoefficientκmaxofτnmaxin(1.6)isnonvanishing,thatis,un=κmaxτnmax+...withκmax6=0.Otherwisethek-tuple{u0,u1,...,uk−1}issaidtobeslowgrowing.LetPk=akyk+ak−1yk−1+...+a0|ai∈C,0≤i≤k denotethelinearspaceofallpolynomialsofdegreeatmostkwithcomplexcoefficients.Definition4.TherealhypersurfaceΞk⊂Pkobtainedastheclosureofthesetofallnondominantpolynomialsiscalledthestandardequimodulardiscriminant.ForanyfamilyΓ(y,z1,...,zq)=ak(z1,...,zq)yk+ak−1(z1,...,zq)yk−1+...+a0(z1,...,zq) RATIONALAPPROXIMATIONOFALGEBRAICFUNCTIONS3ofirreduciblepolynomialsofdegreeatmostkinthevariableywedefinetheinducedequimodulardiscriminantΞΓtobethesetofallparametervalues(z1,...,zq)∈Cqforwhichthecorrespondingpolynomialinyisnondominant.Givenanalgebraicfunctionf(z)definedby(1.1)wedenotebyΞftheinducedequimodulardiscrimi-nantof(1.2)consideredasafamilyofpolynomialsinthevariabley.Example1.Fork=2thee