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arXiv:math/0701567v1[math.CV]21Jan2007PolynˆomedeHua,noyaudeBergmandesdomainesdeCartan-Hartogsetprobl`emedeLuQikengF.ZohraDemmad–Abdessameud∗20janvier2007R´esum´eR´eductionduprobl`emedeLuQikengpourlesdomainesdeCartan-HartogsbΩm(μ)`aunprobl`emealg´ebriquesurlespolynˆomesdeHua.So-lutioncompl`eteduprobl`emedeLuQikengquandledomainedebaseΩestundomainesym´etriquededimensioninf´erieureou´egale`a4.ClassificationAMS(2000):32M15,32A36.Motscl´es:DomainesdeCartan,noyaudeBergman,polynˆomedeHua,conjecturedeLuQikeng.Tabledesmati`eresR´esum´einEnglishlanguage2Introduction71PolynˆomesdutypedeHua92NoyaudeBergmandesdomainesdeCartan-Hartogs113Probl`emedeLuQikengpourlesdomainesdeCartan-Hartogs154Solutionduprobl`emedeLuQikengpourunebasedefaibledimension16ALocalisationdesracines21BTables25∗D´epartementdemath´ematiques,Facult´edesSciences,Universit´eSaadDahlab,RoutedeSoumˆaa,BP270,Blida,Alg´erie;[e-mail],[e-mail]1R´esum´einEnglishlanguageTheLuQikengproblemforadomainU⊂CnconsistsindecidingwhethertheBergmankernelKU(z,w)ofthisdomainmayvanishatsomepointsofU×U.AdomainUiscalledaLuQikengdomainifitsBergmankerneliszero-freeonU×U.LetΩbeanirreducibleboundedcircledhomogeneousdomainandN(z,t)itsgenericnorm.Forμ0andmapositiveinteger,letbΩm(μ)=n(z,Z)∈Ω×Cm,kZk2N(z,z)μo.ThedomainbΩm(μ)iscalledCartan–Hartogsdomain(withbaseΩ,exponentμ,fiberdimensionm).TheBergmankernelofthisdomainmaybeexplicitlycomputedfromthegenericnormandtheHuapolynomialofΩ.If(a,b,r)arethenumericalinvari-ants(multiplicitiesandrank)ofthedomainΩ,itsHuapolynomialisχ(s)=χa,b,r(s)=rYj=1s+1+(j−1)a2
1+b+(r−j)a,where(s+1)k=Qki=1(s+i)denotestheraisingfactorial.ThispolynomialisrelatedtotheHuaintegralbyZΩN(z,z)sω(z)=χ(0)χ(s)ZΩω(Res−1)(cf.[5]).Thedecompositionχ(kμ)=dXj=0μjCd−j(μ)(k+1)jofχ(kμ)alongraisingfactorialsw.r.tokdefinespolynomialsCj(μ),whichareofdegreejinμ.Formapositiveintegerandμ0,definePmμ(η)=dXj=0(m+1)jCd−j(μ)μjηj.Notethatthedegreeofthispolynomialw.r.toηorμisequaltothedimensiondofΩ.TheBergmankernelofbΩm(μ)isthen(cf.[5],[6])bKm,μ((z,Z),(w,W))=CN(z,w)g+mμηm+1Pmμ(η),whereξ,η:bΩm(μ)×bΩm(μ)→Caredefinedbyξ((z,Z),(w,W))=hZ,WiN(z,w)μ,η=11−ξ.2TherangeofξistheunitdiscΔ⊂Candtherangeofηisthehalf-planeReη12 .ThustheLuQikengproblemforbΩm(μ)isreducedtothelocalizationoftherootsofPmμ:Theorem.ThedomainbΩm(μ)isaLuQikengdomainifandonlyifallrootsofPmμarelocatedinReη≤12 .Applyingthistheorem,theLuQikengproblemiscompletelysolvedinthispaperforallm,μ0whenthebaseisofdimensiond≤4.ThisprovidesalotofexamplesofLuQikengandnon-LuQikengdomains.Incontrastwiththegenericcaseofaboundeddomain(“TheLuQikengconjecturefailsgenerically”,see[3]),“most”ofthedomainsbΩm(μ)areLuQikengdomains.Actually,thedomainbΩm(μ)isaLuQikengdomainform≥mΩandforallμ0,wheremΩisanintegerdependingonthebaseΩ;for1≤mmΩ,thereexistsapositiverealnumberμmsuchthatthedomainbΩm(μ)isaLuQikengdomainifandonly0μ≤μm.Resultsareasfollows:1.IfΩistheunitdiscΔ⊂C,thedomainbΩm(μ)isaLuQikengdomainforallm≥1andallμ0.Thisisrecalledhereonlyforsakeofcompleteness.2.IfΩistheHermitianballofdimension2,thedomainbΩm(μ)isaLuQikengdomainifandonlyif1.m=1,μ≤2;2.m=2,μ≤4;3.m≥mΩ=3,forallμ0.Form=1,theresultisduetoH.P.Boas,SiqiFu,E.Straube[4];seealso[7].Form1,resultsarenew.3.IfΩistheHermitianballofdimension3,for1≤m≤5,thepolynomialqm(ofdegree2or3)definedbyqm(μ)=Pmμ12
hasauniquepositiverootμmand0μ1=√2μ2μ3μ4μ5.ThedomainbΩm(μ)isaLuQikengdomainifandonlyif1.1≤m≤5,0μ≤μm;2.m≥mΩ=6,forallμ0.3Form=1,thisresulthasbeenobtainedbyWeipingYin[7],byaslightlydifferentmethod.Resultsarenewform1.4.IfΩistheLieballofdimension3(domainoftypeIV3≃III2,symmetricmatrices),thesametypeofresultholdsasintheprecedingcase,withdifferentqmandμm,0μ1=2√3μ2μ3μ4μ5,butagainmΩ=6.Heretheresultsarenewforallm.5.IfΩistheHermitianballofdimension4,thepolynomialqmdefinedbyqm(μ)=Pmμ12
•hastwopositiverootsμm=μm,1μm,2form=1,2;•hasonepositiverootμmfor3≤m≤7;•ispositiveforallμ0ifm≥mΩ=8.Moreover,0μ1=q32μ2μ3μ4μ5μ6μ7.ThedomainbΩm(μ)isaLuQikengdomainifandonlyif1.1≤m≤7,0μ≤μm;2.m≥mΩ=8,forallμ0.Form=1,thisresulthasbeenobtainedbyJong-doParkandLiyouZhang(2006,unpublished).Resultsarenewform1.6.IfΩistheLieballofdimension4(domainoftypeIV4≃I2,2,2×2matrices),thesametypeofresultasintheprecedingcaseholds,withdifferentqmandμm,0μ1=12p23−√337μ2μ3μ4μ5μ6μ7,andmΩ=8.Heretheresultsarenewforallm.Resultsmaybesummarizedinthefollowingtheorem:Theorem.LetΩbeanirreducibleboundedcircledhomogeneousdomainofdi-mensionatmost4.Thenthepolynomialqm(μ)=Pmμ12
has0,1or2positiveroots.Ifqmhasnopositiveroot,letμm=+∞;ifqmhasonepositiveroot,denotethisrootbyμm=μm,1andletμm,2=+∞;ifqmhastwopositiveroots,denotetheserootsbyμm,1,μm,2andletμm=μm,1μm,2.ThepolynomialPmμhas4•norootinReη12 if0μ≤μm;•onerootinReη12 ifμm,1μ≤μm,2;•tworootsinReη12 ifμm,2μ.TheCartan–HartogsdomainbΩm(μ)isaLuQikengdomainifandonlyif0μ≤μm.Thevaluesofthepositiverootsofqmaregiveninthefollowingtable.TypeI1,2I1,3≃II3III2≃IV3I1,4I2,2≃IV4μ1,12√22√3q32≃1.07732μ1,2+∞+∞+∞4≃3.21549μ2,141+√723+√738≃1.41518≃1.21176μ2,2+∞+∞+∞≃11.333≃9.08062μ3+∞1+q522≃1.61819≃1.41824μ4+