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首页 Notions of Positivity and the Geometry of Polyno…

Notions of Positivity and the Geometry of Polynomials, Branden, Birkhauser, 2011.pdf

Notions of Positivity and the G…

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2011-09-15 0人阅读 举报 0 0 暂无简介

简介:本文档为《Notions of Positivity and the Geometry of Polynomials, Branden, Birkhauser, 2011pdf》,可适用于高等教育领域

TrendsinMathematicsTrendsinMathematicsisaseriesdevotedtothepublicationofvolumesarisingfromconferencesandlectureseriesfocusingonaparticulartopicfromanyareaofmathematicsItsaimistomakecurrentdevelopmentsavailabletothecommunityasrapidlyaspossiblewithoutcompromisetoqualityandtoarchivetheseforreferenceProposalsforvolumescanbesubmittedusingtheOnlineBookProjectSubmissionFormatourwebsitewwwbirkhausersciencecomMaterialsubmittedforpublicationmustbescreenedandpreparedasfollows:Allcontributionsshouldundergoareviewingprocesssimilartothatcarriedoutbyjournalsandbecheckedforcorrectuseoflanguagewhich,asarule,isEnglishArticleswithoutproofs,orwhichdonotcontainanysignificantlynewresults,shouldberejectedHighqualitysurveypapers,however,arewelcomeWeexpecttheorganizerstodelivermanuscriptsinaformthatisessentiallyreadyfordirectreproductionAnyversionofTEXisacceptable,buttheentirecollectionoffilesmustbeinoneparticulardialectofTEXandunifiedaccordingtosimpleinstructionsavailablefromBirkhäuserFurthermore,inordertoguaranteethetimelyappearanceoftheproceedingsitisessentialthatthefinalversionoftheentirematerialbesubmittednolaterthanoneyearaftertheconferencetheGeometryofPolynomialsNotionsofPositivityandPetterBrändénMikaelPassareEditorsMihaiPutinarPrintedonacidfreepaperSpringerBaselAGispartofSpringerScienceBusinessMediawwwbirkhausersciencecomISBN©SpringerBaselAGEditorsDepartmentofMathematicseISBNMathematicsSubjectClassification:A,B,C,E,P,D,P,M,N,P,Q,DOIPetterBrändénDepartmentofMathematicsRoyalInstituteofTechnologyStockholmSwedenpbrandenkthseMikaelPassareStockholmUniversityStockholmSwedenpassaremathsuseDepartmentofMathematicsMihaiPutinarUniversityofCaliforniaatSantaBarbaraSantaBarbara,CAUSAmputinarmathucsbeduLibraryofCongressControlNumber:A,R,B,D,D,E,A,B,A,B,C,M,A,C,A,A,B,A,M,D,K,BownermustbeobtainedThisworkissubjecttocopyrightAllrightsarereserved,whetherthewholeorpartofthematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,broadcasting,reproductiononmicrofilmsorinotherways,andstorageindatabanksForanykindofusepermissionofthecopyrightContentsIntroductionviiSecondecompositiondemathe´matiquesxiSecondCompositioninMathematicsxvPublicationsofJuliusBorceaxixAAlemanandASarafoleanuCommutingLinearDifferentialOperatorsandHankelMatricesMAnderssonandEWulcanVariantsoftheEffectivestellensatzandResidueCalculusBBerndtssonAnExtensionProblemforConvexFunctionsJEBjo¨rk,JBorceaandRBøgvadSubharmonicConfigurationsandAlgebraicCauchyTransformsofProbabilityMeasuresPBra¨nde´n,JHaglund,MVisontaiandDGWagnerProofoftheMonotoneColumnPermanentConjectureMBresˇarandIKlepTracialstellensa¨tzeGCsordasIteratedTura´nInequalitiesandaConjectureofPBra¨nde´nKDeschoutandABJKuijlaarsDoubleScalingLimitforModifiedJacobiAngelescoPolynomialsPDurenandHSShapiroConstrainedApproximationviaFunctionalAnalysisAEremenkoandAGabrielovAnElementaryProofoftheBandMShapiroConjectureforRationalFunctionsviContentsSFriedlandandUNPeledThePressure,DensitiesandFirstorderPhaseTransitionsAssociatedwithMultidimensionalSOFTMGekhtmanandOKorovnichenkoMatrixWeylFunctionsandNonAbelianCoxeterTodaLatticesBGustafssonandVTkachevOntheExponentialTransformofLemniscatesMIsmailandPSimeonovOnaFamilyofPositiveLinearIntegralOperatorsCRJohnson,CMarijua´n,MPisoneroandOWalchMonomialInequalitiesforNewtonCoefficientsandDeterminantalInequalitiesforpNewtonMatricesDKhavinson,RPereira,MPutinar,EBSaffandSShimorinBorcea’sVarianceConjecturesontheCriticalPointsofPolynomialsTMLiggettandAVandenbergRodesStabilityon{,,,}푆:BirthDeathChainsandParticleSystemsKRanestadandBSturmfelsTheConvexHullofaVarietyBReznickBlendersBShiffmanandSZelditchRandomComplexFewnomials,ISTyurinaandAVarchenkoFiniteorderInvariantsfor(푛,)TorusKnotsandtheCurve푌=푋푋Iˆnnoapteundevamaietotceafosts¸inumaie,cesamutat,cesapierdutdintimpulviuıˆntimpulmutIˆnHadese–totceatrecutDinaheronticult¸inutvintoateamintirileIˆnHadese–totceatrecutprieriis¸iiubirile(Innoapteundevamaie,deLucianBlaga)Lostinthenight,somewhere,thereisallthatoncewasandnomoreis,whatgotlost,whatwasuprooted,fromlivingtimetotimethat’smutedInHadesis–allthathaspassedFromAcheron,therivervast,allmemoriestousreturnInHadesis–allthathaspassedthespringtimes,andthelovesweyearn(Lostinthenight,somewhere,thereisbyLucianBlaga)IntroductionThisbookisdedicatedtothememoryofJuliusBorcea,visionarymathematicianwhosefervorandintensityleftnooneindifferentJulius’restlessandconqueringmindcouldnotconcealatormentedheartwhichburneditselfoutalltoquicklytotheconsternationofallthosewhoknewhimHismathematicallegacyconsistsofsingularresults,novelapproachestooldproblemsandaconstellationofopenquestionsA“symphonyofconjectures”,ashelikedtosayThearticlesboundinthepresentvolumewerecommissionedandwrittenwiththeaimofexhibitingandhonoringthedepthofJulius’creativepowersJuliusBogdanBorceawasborninBacaˇu,Romania,inAttheageofhefollowedhisparentstoMoroccothentoDenmarkwherehecompletedhisBaccalaure´atattheLyce´eFranc¸aisofCopenhagenIn–heattendedtheprestigiousLyce´eLouisleGrandinParisHecompletedhismathematicalstudiesinLundunderthesupervisionofArneMeurmanAfterdefendinghisPhDthesisin,heembarkedinpostdoctoralstudiesattheMittagLefflerInstituteforsixmonthsandinStrasbourgfortwoyearsJuliuswasappointedAssistantfromLinguisticTreasonARomanianEnglishpoetrytranslationblogviiiIntroductionProfessoratStockholmUniversityinandLecturerinAyearlaterhewasgrantedtheprestigiousSwedishMathematicalSocietyWallenbergPrizePromotedFullProfessorin,hewasawardedthemostdistinguishedRoyalAcademyofSciencesFellowshipinHediedonApril,JuliuswasaprominentresearcherandadedicatedmentorandteacherHisscientificworkrangesfromvertexoperatortheorytozerodistributionofpolynomialsandentirefunctions,viacorrelationinequalitiesandstatisticalmechanicsJulius’thesisalreadyborethemarkofhisoriginalityItconsistsoftwoseeminglyindependentparts:oneinvertexoperatortheoryandtheotherdevotedtothegeometryofzerosofcomplexpolynomialsinonevariableInvertexoperatortheoryJuliusgeneralizedresultsofPrimcandMeurmanandgaveaclassificationofannihilatedfieldsAsconcernscomplexpolynomials,hetackledSendov’sconjectureonzerosandcriticalpointsofcomplexpolynomialsinonevariableUsingnoveltechniques,heprovedtheconjectureforpolynomialsofdegreenotexceedingEarlier()theconjecturehadbeenprovenforpolynomialsofdegreenotexceedingAtStockholmUniversityJuliushadasteadycollaborationwithBøgvadandBorisShapiroTheyworkedonrationalapproximationsofalgebraicequations,piecewiseharmonicfunctionsandpositiveCauchytransforms,andthegeometryofzerosofpolynomialsinonevariableBorceaandBra¨nde´ncollaboratedonaprojectonthegeometryofzerosofpolynomialsandentirefunctionsTheycharacterizedalllinearoperatorsonpolynomialspreservingthepropertyofhavingonlyrealzeros,aproblemthatgoesbacktoLaguerreandPo´lya–SchurTheseresultsweresubsequentlyextendedtoseveralvariables,andaconnectiontotheLee–YangprogramonphasetransitionsinstatisticalphysicswasmadeTogetherwithTomLiggett(UCLA)theyappliedtheirmethodstoproblemsinprobabilitytheoryandwereabletoproveanimportantconjectureaboutthepreservationofnegativedependencepropertiesinthesymmetricexclusionprocessJuliushadacomprehensiveprojectonthedistributionofpositivechargesandtheHaussdorffgeometryofcomplexpolynomialsOneofthemotivationsfortheprojectwastobringSendov’sconjectureintoalargerandmorenaturalcontextHeformulatedseveralinterestingconjectures,andinthesummerofhewasthedrivingforceoftwomeetings,oneattheAmericanInstituteofMathematicsinPaloAltoandtheotherattheBanffInternationalResearchStationtogetherwithKhavinson,Pereira,Putinar,SaffandShimorinThesetwoencounterswerefocusedonstructuringandexpandingJulius’programHiscontinuousandvividinterestintheHausdorffgeometryofpolynomialswastriggeredbyanE´coleNormaleSuperieure(Paris)examhetookinJuliushadalivelyinterestinliteratureAsayoungman,hewrotepoetryandhaddreamsofbecomingawriter,butinhisteenyears,hisinterestformathematicsbecameprevalentJuliuslivedformathematicsHehadacomplexpersonalityWethankOliverDebarreandGuyHenniartforgrantinguspermissiontoreproducebelowtheoriginalexamsheetIntroductionixJuliusBogdanBorcea(–)xIntroductionCurious,passionateandturbulent,butalsosensitive,caring,generous,andfirstandforemostintenseineverythinghedidHeissurvivedbyhiswifeRoxana(whomhemarriedinPraguein),hismother(todayretiredinRomania),andhisbrother(settledwithhisfamilyinMalmo¨)HisfatherdiedinMalmo¨oneyearafterhimThegapleftinthemathematicalcommunitybyhisdeathespeciallyamongthosewhowereprivilegedtoknowandinteractwithhimisincommensurableInthewakeofhisincomprehensibledisappearance,wecarryandwillstrivetopromotehisvisionandinvaluablemathematicallegacyWeareindebtedtoThomasHempfling,ManagingEditoratSpringerandExecutiveEditoratBirkha¨userforhisenthusiasticandconstantsupportWeexpressourgratitudetoallcontributorstothepresentvolumeWealsothankChristineChodkiewiczPutinarforhernonmathematicalencouragements,andRoxanaBorceaforherhelpwiththeintroductionandforprovidingthephotographofJuliusPetterBra¨nde´nMikaelPassareMihaiPutinarxiiENS:Secondecompositiondemathe´matiquesENS:Secondecompositiondemathe´matiquesxiiiSecondCompositioninMathematicsDuration:hoursThefollowingpagesareatranslationoftheexaminationsubjectJuliusBorceatookinThetopicsofthisexam(thegeometryofzerosandcriticalpointsofcomplexpolynomials)returnedobsessivelyonhisworkingagendaandshapedagoodportionofhisresearchpathWethankOliverDebarre(E´coleNormaleSuperieure)andGuyHenniart(Universite´ParisSud)forgrantingpermissiontoreproduceinthisvolumetheexaminfulldetailTheaimofthisproblemistoprove,insomeparticularcases,theconjectureexplainedbelow,usuallyknownastheIlievSendovconjectureOnedenotebyℂthefieldofcomplexnumbers,and∣푧∣standsfortheabsolutevalue,ormodulus,ofthecomplexnumber푧onedenotesbyℜ푧itsrealpartWeendowℂwiththestructureofatwodimensionalEuclideanrealvectorspaceassociatedtothenorm∣푧∣WewillassumeasknowntheTheoremduetod’AlembertandGauss,statingthateverynonconstantpolynomialwithcomplexcoefficientshasacomplexrootLet푆∈ℂ푋beapolynomialwithcomplexcoefficients,ofdegreeatgreaterorequalthanLet푧bearootof푆Wesaythat푆and푧satisfy(IS)ifthereexistsaroot휁ofthederivative푆′satisfying푧−휁∣≤Wesaythat푆satisfies(IS)if,foreveryroot푧of푆,thecouple푧,푆satisfies(IS)IlievSendov’sconjectureassertsthateverydegreetwoorhigherpolynomialofℂ푋,havingallrootsinmoduluslessthanorequalto,satisfies(IS)Fromnowonwefixaninteger푛∕=andadegree푛polynomial푃=푎푛푋푛⋅⋅⋅푎∈ℂ푋Wedenoteby푧,,푧푚thedistinctrootsof푃(notethat푚isalsoanonnegativeinteger)for푖=,,푚wedenoteby푛푖themultiplicityoftheroot푧푖Therefore:푃=푎푛푚∏푖=(푋−푎푖)푛푖and푚∑푖=푛푖=푛Weassumethatthe푧′푖푠satisfy∣푧푖∣≤FinallyweremindthatapolynomialwhosehighestdegreetermhasitscoefficientequaltoiscalledamonicpolynomialIAfewsimplecasesoftheconjectureAProvethatif푛=,then푃satisfies(IS)Provethatif푛≥,then푃and푧satisfy(IS)xviSecondCompositioninMathematicsProvethattherearecomplexnumbers푤,,푤푚whicharenotrootsof푃,suchthat:푃′=푛푎푛푚∏푖=(푋−푧푖)푛푖−푚∏푗=(푋−푤푗)HenceforthwepreserveinSectionIAthenotationintroducedintheaboveformulaAssume푛=Compute∏푚푗=(푧−푤푗)asafunctionof푛and푧푖”푠Provethat푃verifies(IS)whenever푛≥푚Decompose푃′푃insimplefractionsLet푗∈{,,푚}Show,considering(푃′푃)(푤푗),that푤푗isabarycenterwithstrictlypositivecoefficientsofthe푧′푖푠andthat∣푤푗∣≤Deducethat,if푧=,then푃and푧verify(IS)BLetusslightlychangethenotation:푃′=푛푎푛∏푛−푖=(푋−푡푖),where푡푖arecomplexnumbersInthissectionIBweassume푛=Showthat∣(푃′′푃′)(푧)∣≥푛−implies푃and푧satisfy(IS)Compute(푃′′푃′)(푧)asafunctionofthe푧′푖푠usingthepolynomial푃(푋−푧)Provethat,if푧∈ℂsatisfies∣푧∣≤,푧∕=,thenℜ(−푧)≥Weassume푧=andwearrangethe푡′푖푠sothatℜ(−푡)≥ℜ(−푡푖)for푖=,,푛−Showthatℜ(−푡)≥then∣푡−∣≤and∣푡−∣≤Assumethat푧hasmodulusThenprovethat푃and푧satisfy(IS)UseasimplegeometrictransformofℂIIThecaseofarealrootThroughoutthissectionweassume푛=and푧isarealnumber푎satisfying<푎<For푤∈ℂ∖{푎}weset푇(푤)=푤−푎푤−Wedenoteby푃˜thepolynomialofℂ푋satisfying:푃˜(푋)=(푎푋−)푛푃(푋−푎푎푋−),writing푃˜(푋)=푏푛푋푛⋅⋅⋅푏,푏푖∈ℂCompute푇∘푇(푤)for푤∈ℂ∖{푎}andfindtheimageoftheunitcircle,itsinteriorandexteriorminusthepoint푎SecondCompositioninMathematicsxviiProvethat푏=,∣푏∣≤∣푏푛∣and∣푏푛−≤(푛−)∣푏푛∣Wedefine푅(푋)=∑푖=푛(푛−푖)푏푖푋푖푖푏푖푎푋푖−andwewrite푅(푋)=퐴∏푛−푘=(푋−훾푘),where퐴isnotzeroandthe훾′푘푠arearrangedsothat∣훾∣≤∣훾∣≤⋅⋅⋅≤∣훾푛−∣Provethat∏푛−푘=∣훾푘∣≤푛−푎(푛−)Let푤∈ℂ∖{푎}Compute푃′(푇(푤))asafunctionof푎,푤and푅(푤)Let휇bearealnumbersatisfying∣훾∣≤휇≤푎Provethat푃′admitsaroot휁withtheproperty:∣휁−푎∣≤휇(−푎)−푎휇If휇satisfies휇≤푎−푎,thenprovethat푃′hasaroot휁with∣휁−푎∣≤Provethatinthecase푛≤,푃and푎satisfy(IS),anddeducethat,for푛≤,andanyroot푧of푃whosemodulusisstrictlybetweenand,푃and푧satisfy(IS)Provethatanypolynomialinℂ푋,ofdegreeor,withrootsinmodulusatmostoneverifies(IS)Weassumethat푛=,orandthat푃admitsatleastonemultiplerootofmodulusProvethat푃and푎satisfy(IS)Onecanstudythefunction(푛−)ln(푎−푎)−ln(푛−(푛−)푎)Provethateverypolynomialbelongingtoℂ푥,ofdegree,or,possessingatleastamultiplerootofmodulus,andwithallrootsofmoduluslessorequalthan,verifies(IS)IIIContinuityoftherootsofapolynomialWedenotebyℂ푋푛thevectorspaceofpolynomialsbelongingtoℂ푋ofdegreeatmost푛For푆=푛∑푖=푠푖푋푖∈ℂ푋푛,onedenotes∣푆∣=∑푖=푛∣푠푖∣Thisisanormonℂ푋푛Provethat,if푆∈ℂ푋hasdegree푛,theneverycomplexroot푧of푆satisfies∣푧∣≤∣푆∣∣푠푛∣Let푆푘beasequenceofelementsofℂ푋푛convergingto푆(when푘tendstoinfinity)Denote푆푘=훼푘∏푛푖=(푋−푥푖,푘)Let푧bearootof푆ofmultiplicity푝Provethat,foreveryrealnumber휖>thereexistsanℓsothatatleast푝ofthecomplexnumbers푥푖,푘(푖=,,푛)areatdistanceatmost휖from푧,forall푘≥ℓxviiiSecondCompositioninMathematicsIVExtremalpolynomialsLet푘beanintegersatisfying푛≥푘≥Let푃푛(푘)denotethesetofmonicpolynomialsbelongingtoℂ푋푛andpossessingatmost푘distinctroots,allofmoduluslessthanorequaltoFor푆∈푃푛(푘)andforaroot푧of푆,denoteby퐼푆(푧)theshortestdistancefrom푧totherootsof푆′Denoteby퐼(푆)themaximumamong퐼푆(푧)when푧runsoverallrootsof푆AProvethat퐼(푆)≤for푆∈푃푛(푘)andthatthereexistsapolynomial푆∈푃푛(푛−)suchthat퐼(푆)=Wedenoteby퐼(푃푛(푘))theupperboundof퐼(푆)when푆belongsto푃푛(푘)Showthat,if퐼(푃푛(푘))≤,theneveryelementof푃푛(푘)verifies(IS)Provethat푃푛(푘)isacompactsubsetofℂ푋푛Provethat퐼:푆→퐼(푆)isacontinuousmapfrom푃푛(푘)intoℝandthatthereexistsapolynomial푆∈푃푛(푘)suchthat퐼(푆)=퐼(푃푛(푘))BAnelement푆∈푃푛(푘)iscalledextremalif퐼(푆)=퐼(푃푛(푘))Provethatanextremalpolynomialin푃푛(푘)hasarootofmodulusoneShowthat,foreveryrealnumber휃,anextremalpolynomialin푃푛(푘)hasatleastonerootoftheform푒푖훼,where훼∈휃,휃휋)Weassume푛=,orand푘=Let푆beanextremalelementof푃푛(푘)andassumethat푆admitsarealroot푎satisfying<푎<Provethat푆and푎verify(IS)Incase

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Notions of Positivity and the Geometry of Polynomials, Branden, Birkhauser, 2011

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