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On Level Curves and Conformal Equivalence of Meromorphic Functions

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Title:
On Level Curves and Conformal Equivalence of Meromorphic Functions
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1 online resource (112 p.)
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english
Creator:
Richards, Trevor J
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University of Florida
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Gainesville, Fla.
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Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Mathematics
Committee Chair:
Jury, Michael Thomas
Committee Members:
Cenzer, Douglas A
Shen, Li C
Mccullough, Scott A
Hobert, James P

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Subjects / Keywords:
complex-analysis -- level-set -- polynomial
Mathematics -- Dissertations, Academic -- UF
Genre:
Mathematics thesis, Ph.D.
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theses   ( marcgt )
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Abstract:
In this dissertation we study the level curves of the class of meromorphic functions $(f,G)$ which are conformally equivalent to a finite Blaschke product on the disk, and, in particular, conformal equivalence of these functions in light of their respective level curve structures. We begin with a general study, and then apply our findings to the study of the equivalence classes of such functions modulo conformal equivalence. We develop the notion of a level curve structure, and show that conformal equivalence of the meromorphic functions referred to above may be determined entirely in terms of level curve structure.
General Note:
In the series University of Florida Digital Collections.
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Includes vita.
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Includes bibliographical references.
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Description based on online resource; title from PDF title page.
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This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility:
by Trevor J Richards.
Thesis:
Thesis (Ph.D.)--University of Florida, 2013.
Local:
Adviser: Jury, Michael Thomas.

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MISSING IMAGE

Material Information

Title:
On Level Curves and Conformal Equivalence of Meromorphic Functions
Physical Description:
1 online resource (112 p.)
Language:
english
Creator:
Richards, Trevor J
Publisher:
University of Florida
Place of Publication:
Gainesville, Fla.
Publication Date:

Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Mathematics
Committee Chair:
Jury, Michael Thomas
Committee Members:
Cenzer, Douglas A
Shen, Li C
Mccullough, Scott A
Hobert, James P

Subjects

Subjects / Keywords:
complex-analysis -- level-set -- polynomial
Mathematics -- Dissertations, Academic -- UF
Genre:
Mathematics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract:
In this dissertation we study the level curves of the class of meromorphic functions $(f,G)$ which are conformally equivalent to a finite Blaschke product on the disk, and, in particular, conformal equivalence of these functions in light of their respective level curve structures. We begin with a general study, and then apply our findings to the study of the equivalence classes of such functions modulo conformal equivalence. We develop the notion of a level curve structure, and show that conformal equivalence of the meromorphic functions referred to above may be determined entirely in terms of level curve structure.
General Note:
In the series University of Florida Digital Collections.
General Note:
Includes vita.
Bibliography:
Includes bibliographical references.
Source of Description:
Description based on online resource; title from PDF title page.
Source of Description:
This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility:
by Trevor J Richards.
Thesis:
Thesis (Ph.D.)--University of Florida, 2013.
Local:
Adviser: Jury, Michael Thomas.

Record Information

Source Institution:
UFRGP
Rights Management:
Applicable rights reserved.
Classification:
lcc - LD1780 2013
System ID:
UFE0045338:00001


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ONLEVELCURVESANDCONFORMALEQUIVALENCEOFMEROMORPHICFUNCTIONSByTREVORJ.RICHARDSADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2013

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c2013TrevorJ.Richards 2

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TomylittlebabyAbigail,thatImaygetajobandbuyyoufood 3

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ACKNOWLEDGMENTS ThankstomygraduateadvisorDr.MichaelJuryandthemembersofmygraduatecommittee.ThanksespeciallytoDr.StephenSummersandDr.Li-ChienShenforhelpingmeasIrstbeganmyresearch. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFFIGURES ..................................... 6 ABSTRACT ......................................... 7 CHAPTER 1HISTORYANDOVERVIEW ............................. 8 2PRELIMINARIES ................................... 13 2.1LevelCurvesasPlanarGraphs ........................ 13 2.2AssortedPropertiesofLevelCurves ..................... 17 2.2.1Setting .................................. 17 2.2.2Properties ................................ 17 3THEPOSSIBLELEVELCURVECONFIGURATIONSOFAMEROMORPHICFUNCTION ...................................... 31 3.1ConstructionofPC ............................... 31 3.2Constructionof ................................ 36 4RESPECTSCONFORMALEQUIVALENCE .................. 40 5ISSURJECTIVE:THEGENERICCASE .................... 44 6ISSURJECTIVE:THEGENERALCASE .................... 57 APPENDIX ASEVERALRESULTS ................................ 88 BSEVERALLEMMATA ................................ 95 REFERENCES ....................................... 111 BIOGRAPHICALSKETCH ................................ 112 5

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LISTOFFIGURES Figure page 2-1AdmissibleGraphs .................................. 29 2-2Non-AdmissibleGraphs ............................... 29 2-3f(z)=z5)]TJ /F4 11.955 Tf 11.96 0 Td[(1 ..................................... 29 2-4D,z1,andz2 ..................................... 30 2-5Denitionof ..................................... 30 5-1Tractoff ....................................... 56 5-2Tractofp ....................................... 56 A-1Gauss'Theorem ................................... 94 6

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyONLEVELCURVESANDCONFORMALEQUIVALENCEOFMEROMORPHICFUNCTIONSByTrevorJ.RichardsAugust2013Chair:JeanLarsonMajor:MathematicsInthisdissertationwestudythelevelcurvesoftheclassofmeromorphicfunctions(f,G)whichareconformallyequivalenttoaniteBlaschkeproductonthedisk,and,inparticular,conformalequivalenceofthesefunctionsinlightoftheirrespectivelevelcurvestructures.Webeginwithageneralstudy,andthenapplyourndingstothestudyoftheequivalenceclassesofsuchfunctionsmoduloconformalequivalence.Wedevelopthenotionofalevelcurvestructure,andshowthatconformalequivalenceofthemeromorphicfunctionsreferredtoabovemaybedeterminedentirelyintermsoflevelcurvestructure. 7

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CHAPTER1HISTORYANDOVERVIEWThestudyofthelevelsetsofameromorphicfunctionf(setsoftheformfz:jf(z)j=g)ortractsofthefunction(setsoftheformfz:jf(z)j
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Finally,in1986,K.Stephenson[ 8 ]provedthefollowinggeneralresultwhichimpliesmuchoftheearlierwork.HereifWisasimplyconnectedRiemannsurface,and)]TJ /F1 11.955 Tf 10.09 0 Td[(isanarcinW,thenweletF)]TJ /F1 11.955 Tf 8.64 1.79 Td[(denotethecollectionofallnon-constantmeromorphicfunctionsonWhavingmodulusoneon)]TJ /F1 11.955 Tf 6.78 0 Td[(. Theorem1.3(LevelCurveStructureTheorem). LetWbeasimplyconnectedRiemannsurface,)]TJ /F9 11.955 Tf 10.1 0 Td[(anarconW,andsupposeF)]TJ /F9 11.955 Tf 8.64 1.8 Td[(isnonempty.ThenthereexistsauniquesimplyconnectedsurfaceS(oneofC,cl(C),orD)andananalyticfunction:W!SsuchthatF)]TJ /F2 11.955 Tf 8.64 1.79 Td[(FRfF:2FRg.Thereisasomewhatdifferentclassofquestionswhichhasarisenandwhichconcernssmoothshapesingeneral,butinwhichthelevelcurvesofmeromorphic(andinparticularrationalandpolynomial)functionsplayalargepart.ThesehavetodowithangerprintwhichasmoothcurveimposesontheunitcircleT,whichwasintroducedbyA.A.Kirillovin1987[ 9 10 ].Let)]TJ /F1 11.955 Tf 10.1 0 Td[(beasmoothsimpleclosedcurveinC,withboundedface)]TJ /F1 11.955 Tf 10.4 1.79 Td[(andunboundedface+.Let)]TJ /F4 11.955 Tf 7.08 1.79 Td[(,+denoteRiemannmapsfromD,D+to)]TJ /F4 11.955 Tf 7.08 1.8 Td[(,+respectively(hereD+isdenedasCncl(D)).WithcertainnormalizationsontheReimannmaps,wedenethengerprintkof)]TJ /F1 11.955 Tf 10.1 0 Td[(byk:=+)]TJ /F5 7.97 Tf 6.58 0 Td[(1)]TJ /F1 11.955 Tf 7.08 1.8 Td[(.Since)]TJ /F1 11.955 Tf 10.1 0 Td[(issmoothitiseasytoshowthatkisadiffeomorphismfromTtoT.Moreoverif^)]TJ /F1 11.955 Tf 10.09 0 Td[(equalstheimageof)]TJ /F1 11.955 Tf 10.1 0 Td[(underanafnetransformationf(z)=az+b,withcorrespondingngerprint^k,thenk=^kforsomeautomorphism:D!D.ThereforewemaydeneafunctionFwhichmapsasmoothsimpleclosedcurves(modulocompositionwithafnetransformation)tothecorrespondingdiffeomorphismofTwhichisitsngerprint(modulopre-compositionwithanautomorphismofD).(Note:Thisandmorebackgroundmaybefoundin[ 11 ].)Kirillovprovedthefollowingtheorem[ 9 10 ]. Theorem1.4. Fisabijection.Ifwerestrictourattentiontosmoothcurveswhichariseaslevelcurvesofpolynomials,asimilarresultmaybeobtained.Onerstshowsthatif)]TJ /F1 11.955 Tf 10.1 0 Td[(isaproperpolynomiallemniscate(ie)-297(=fz:jp(z)j=1gforsomen-degreepolynomialpsuchthat 9

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fz:jp(z)j=1gissmoothandconnected)thenthecorrespondingngerprintk=B1 nforsomen-degreeBlaschkeproduct.ThenifweletFpdenotethefunctionFviewedashavingasitsdomaintheequivalenceclassesofsimplesmoothclosedcurveswhichariseasproperpolynomiallemniscates,andhavingcodomaintheequivalenceclassesofdiffeomorphismsofTconsistingofnthrootsofn-degreeBlaschkeproducts(n2N),thenonemayprovethefollowingtheorem. Theorem1.5. Fpisabijection.Thisresultwasstatedin[ 11 ]andfollowsratherdirectlyfromTheorem 1.4 ,thoughtheauthorsproveditusingothermeans1.InChapter 5 andChapter6,wewillproveasomewhatstrongerresultwhichhasTheorem 1.5 asacorollary,thoughinthefollowingequivalentform. Theorem1.6. ForeveryniteBlaschkeproductBwithdegreen,thereissomendegreepolynomialpsuchthatthesetG:=fz:jp(z)j<1gisconnected,andsomeconformalmap:D!GsuchthatB=ponD.Ourmaingoalinthispaper,however,istoexplorethewayinwhichthecongurationoflevelcurvesofameromorphicfunctioncharacterizethatfunctionmoduloconformalequivalence.InChapter 2 webuildupseveralpreliminarypropertiesofthelevelcurvesofameromorphicfunction.OursettingwillbeameromorphicfunctionfwithaboundednitelyconnecteddomainGsuchthatfismeromorphicacrosstheboundaryofG,jfjisconstantonanycomponentof@G,andf06=0on@G.Inthissetting,wemakethefollowingdenition. Denition1. Ifw2C,andfismeromorphicatw,thenweletwdenotethecomponentoffz2dom(f):jf(z)j=jf(w)jgwhichcontainsw.OneofthemainresultfromthischapterisTheorem 2.1 ,whichstatesthefollowing. Theorem1.7. LeteachofL1andL2bealevelcurveoffcontainedinGoraboundedcomponentofGc.LetF1denotetheunboundedfaceofL1andF2theunboundedface 10

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ofL2.IfL1F2,andL2F1,thenthereissomew2GwhichliesinF1\F2,suchthatf0(w)=0andL1andL2arecontainedindifferentboundedfacesofw.ThisexistencetheoremforcriticalpointsoffisprovedbymeansoftheseparationresultProposition 2.4 ,whichstatesbroadlyspeakingthatifLisalevelcurveoffinG,andKisacompactsetwhichdoesnotintersectL,thenKisseparatedfromLbyanotherlevelcurveoffinG.Theothermainresult(Theorem 2.1 )fromChapter 2 statesinbriefthefollowing. Theorem1.8. Foranymeromorphicfunction(f,G),ifthenitelymanycriticallevelcurvesoffinG(thatisthelevelcurvesoffinGwhichcontaincriticalpointsoff)areremoved,thenineachoftheremainingregionsfisconformallyequivalenttoapurepowerofz.Morespecically,ifDisoneoftheremainingregionsthenthereisanannulusAandaconformalmap:D!Aandsomen2Znf0gsuchthatfnonD.Thismaybeviewedasthenaturalextensionofthefactthatifwisazeroorpoleofanymeromorphicfunctiong,thenthereisaneighborhoodofwonwhichgisconformallyequivalenttoapurepowerofz.TheproofofthistheoremreliesheavilyonTheorem 2.1 .InChapter 3 ,wemakethefurtherassumptiononourfunctions(f,G)thatGissimplyconnected,andwerigorouslyconstructasetPCwhichrepresentsallpossiblecongurationsofcriticallevelcurvesof(f,G).Wethendeneafunctionwhichmaps(f,G)tothecorrespondingcongurationinPC.Chapter 4 containsthemainresult(Theorem 4.1 )ofthepaper,namelythatrespectsconformalequivalence. Theorem1.9. If(f1,G1)and(f2,G2)aretwofunctionsasdescribedabove,then(f1,G1)(f2,G2)ifandonlyif(f1,G1)=(f2,G2).Here(f1,G1)(f2,G2)meansthatthereissomeconformalmap:G1!G2suchthatf1=f2onG1.Thisresultimpliesthatifweviewashavingforitsdomainthesetofequivalenceclassesofmeromorphicfunctionsmoduloconformalequivalence,thenrstiswelldened,andsecondisinjective. 11

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InChapter 5 andChapter 6 ,weshowthatinalimitedsense,issurjective.Thatis,wedeneasubsetPCaPCofcongurationswhichnaturallycorrespondtoanalyticfunctions.Thenifweviewashavingforitsdomaintheequivalenceclassesofanalytic(f,G)moduloconformalequivalence,andhavingcodomainPCa,thenissurjective.InChapter 5 ,weshowthattheimageofcontainseachcongurationinPCaallofwhosecriticalvaluesarenon-zeroandhavedifferentmoduli.InChapter 6 ,weextendthistoallofPCabyapproximatingageneralmemberofPCabyagenericone.Finally,intherstappendix,wecatalogueseveralfactsaboutlevelcurvesandapplicationstolevelcurveswhichwerenotofdirectuseinprovingthemainresultsofthepaper,buthavesomeindependentinterest,includinganapparentlynewproof,usinglevelcurves,ofthefollowingGauss-Lucastheorem. Theorem1.10. Thecriticalpointsofapolynomialparecontainedintheconvexhullofthezerosofp.Andinthesecondappendixweproveseverallemmatausedthroughoutthepaper. 12

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CHAPTER2PRELIMINARIES 2.1LevelCurvesasPlanarGraphs Denition2. Byplanargraphwemeananitegraphembeddedinaplaneorsphere,whoseedgesdonotcross.Forthepurposesofthispaperwewillincludesimpleclosedpathswithnospeciedverticesasplanargraphs. Denition3. LetGCbeaxedopenset,andletf:G!Cbeanon-constantmeromorphicfunction.Let>0begiven,anddeneEf,:=fz2G:jf(z)j=g.Byanlevelcurveoff,wemeanacomponentofEf,.Letbesomelevelcurveoffwhichisbounded,andsuchthatcl()G.Let>0denotethevaluejfjtakeson.Ingeneral,byalevelcurveoffwemeanacomponentofEf,forsome>0.Foranyz2G,letzdenotethelevelcurveoffwhichcontainsz.Intheexistingliteratureonthesubjectoflevelcurvesofameromorphicfunction,itisgenerallyassumedwithoutproofthatisaplanargraphwhoseverticesarethecriticalpointsoffin,andwhoseedgesarepathsinwithendpointsatthecriticalpointsoff.Howeverbecausepartofthesubjectofthisdissertationispreciselythegeometryofthelevelcurvesofameromorphicfunction,wemakethatfactexplicitinProposition 2.1 ,andgiveanindicationastothenatureoftheproof.Becausemuchoftheproofamountstoanexerciseinanalyticcontinuationandcompactness,weleaveoutmostofthedetails. Proposition2.1. isaplanargraph,whoseverticesarethecriticalpointsoff.Ifdoesnotcontainanycriticalpointoff,thenisasimpleclosedpath. Proof. Sinceisboundedandfisnon-constant,thereareonlynitelymanycriticalpointsoffin.Choosesomez02whichisnotacriticalpointoff.Wewishtoshowthateitherz0liesinapathinwhoseendpointsarecriticalpointsoff,orthatcontainsnocriticalpointsoff,inwhichcaseconsistsofasimpleclosedpath. 13

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Sincef0(z0)6=0,thereissomeopensimplyconnectedsetDGwhichcontainsz0suchthataninversegoffmaybedenedonf(D).f(z0)liesinthecircleofradiusaround0.Ourstrategyistocontinueganalyticallyaroundthatcircleasfaraspossible,andconsiderthepaththatthistracesoutbackinthedomainoff. Case2.0.1. gmaybecontinuedindenitelyaroundthe-circleinatleastonedirection.Assumethatgmaybecontinuedindenitelyaroundthe-circleinthepositivedirection.Bythecompactnessof,thereareonlynitelymanypointsinatwhichftakesthevaluef(z0).Thereforeifgiscontinuedanalyticallyalongthe-circleinthepositivedirectionenoughtimes,thepathtracedoutinthedomainintersectsitself.Letdenotethispath.Becausefisinvertibleateachpointinwemayconcludethatissimple(asfwouldnotbeinjectiveatacrossingpointofsuchapath)anddoesnotcontainanycriticalpointoff.Furthermore,wecanusetheinvertibilityoffateachpointin,andtheconnectednessof,toshowthatthepathisallof.Thusisasimpleclosedpathwhichdoesnotcontainanycriticalpointsoff.Ifgmaybecontinuedindenitelyaroundthe-circleinthenegativedirectionweobtainthesameresult. Case2.0.2. gmaynotbecontinuedindenitelyaroundthe-circleineitherdirection.Againletdenotethepathobtainedbycontinuinggaroundthe-circleasfarasmaybeinthepositivedirection.Clearlyiscontainedinanddoesnotcontainanycriticalpointsoff(becausefisinvertibleateachpointin).Onecanusethecompactnessoftoshowthatapproachesauniquepointin,andonecanshowthatthispointisacriticalpointoff(otherwiseonecouldcontinuegfurtheralongthe-circle).gmaynotbecontinuedindenitelyaroundthecircleintheoppositedirectionbecausethiswouldimplythatdoesnotcontainacriticalpointoff(asseeninCase 2.0.1 above).Thereforeifwenowallowtodenotethepathobtainedbycontinuinggalongthe-circleintheoppositedirection,weobtainanotherpathfromz0toacriticalpointoff.Weconcludethatz0liesinapathinwhoseendpointsare 14

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criticalpointsoff.Furthermorethispathissimplebecausefisinvertibleateachpointinthispathotherthantheendpoints.Finallywemayagainusethecompactnessoftoshowthatcontainsonlynitelymanyedgesandvertices,whichgivesusthedesiredresult. Duetothenicebehaviorofrationalfunctionsnear1,onecaneasilyextendtheresultoftheabovepropositiontoalllevelcurvesofarationalfunction.Sincetheproofmerelyrequiresonetopre-composetherationalfunctionwithaMobiusfunction,weomitithere. Corollary1. Ifg:cl(C)!cl(C)isarationalfunction,everylevelcurveofgisaplanargraph.Wewillnowuncoversomerestrictionsonwhichplanargraphswemayseeaslevelcurvesofourfunctionf.FirstinProposition 2.2 weseethatmusthaveboundedfaces,andthateachedgeofisadjacenttoatleastoneofitsboundedfaces.InProposition 2.3 ,weseethateachvertexofmusthaveevenlymanyedgesofincidenttoit. Proposition2.2. hasoneormoreboundedfacesandasingleunboundedface,andeachedgeofisadjacenttotwodistinctfacesof. Proof. Sinceisbounded,thereisasingleunboundedfaceof,sotoshowthathasboundedfaces,itsufcestoshowthesecondresultoftheproposition,thateachedgeofisadjacenttotwodistinctfacesof.DeneE:=fz2Ef,:z=2g.Thencl(E)isaclosedsetcontainedincl(G),andcl(E)doesnotintersect.Choosesomez02whichisnotazerooff0.Thatis,z0isapointinoneoftheedgesof.Choosesome>0smallenoughsothatB(z0)GnE.(Soforallz2B(z0),ifjf(z)j=,thenz2.)BytheOpenMappingTheorem,therearepointsx,y2B(z0)suchthatjf(x)j.Supposebywayofcontradictionthatxandyareinthesamefaceof.Lemma 1 (with=XandE[Gc=Y)givesthatxandyareinthesamecomponentof([E[Gc)cwhichisequaltoGnEf,.Since 15

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openconnectedsetsinCarepathconnected,thereisapath:[0,1]!GnEf,fromxtoy.Butfiscontinuous,andjf(x)j<
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Hereareseveralexamplesofthesegraphswhichariseaslevelcurvesofanalyticfunctions. Example: Letf(z)=znforsomen2Znf0g,and2(0,1).ThenEf,isthecirclefz2C:jzj=1 ng. Example: Letf(z)=zn)]TJ /F4 11.955 Tf 12.9 0 Td[(1forsomen2f2,3,...g.If2(0,1),thenEf,hasncomponents,eachasimpleclosedpathwhichcontainsasinglezerooffinitsboundedface.If2(1,1),thenEf,consistsofasinglesimpleclosedpathwhichcontainsallnzerosoffinitsboundedface.Finally,Ef,1consistsofasinglecomponentwithasinglevertex(at0),nedges,andnboundedfaces,eachofwhichcontainsasinglezerooff.InFigure 2-3 weseetheexamplef(z)=z5)]TJ /F4 11.955 Tf 12.05 0 Td[(1.ThelevelsetsshownarethesetsEf,.5,Ef,1,andEf,1.5. 2.2AssortedPropertiesofLevelCurves 2.2.1SettingWenowimposethefollowingadditionalrestrictionsonthesetG. Gisbounded. ThereissomeopensetG0suchthatcl(G)G0andfismeromorphiconG0. ForeachcomponentKof@G,thereissomer2(0,1)suchthatjfjronK.(Thatis,eachcomponentof@GiscontainedinsomelevelcurveoffinG0.) Note: IfoneofthecomponentsofGcisasinglepointfzgforsomez2C(andthuszisaremovablediscontinuityoffrestrictedtoG),thenwereplaceGwithG[fzg. AnylevelcurveoffinG0thatcontainspartoftheboundaryofGdoesnotextendintoG. GisconnectedandGchasnitelymanycomponents. 2.2.2PropertiesIntheprevioussectionweexaminedonelevelcurveoffinisolationfromtheothers.InthissectionwecataloguesomefactsaboutthelevelcurvesoffgloballyinG.Wewillbeginbyintroducingsomevocabulary. 17

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Denition5. Alevelcurveoffwhichisnotazeroorpoleoff,andcontainsacriticalpointoff,wewillcallacriticallevelcurveoff.Alevelcurveoffwhichisnotacriticallevelcurvewecallanon-criticallevelcurveoff.Inthenextresultweshowthataclosedsetcontainedinthecomplementofalevelcurveoffmaybeseparatedfromthatlevelcurvebyadifferentnon-criticallevelcurveoff.Thisbeginstoshowthatthelevelsetsoffvarycontinuously. Denition6. ForDC,letsc(D)denotetheunionofDwitheachboundedcomponentofDc.(Thescstandsforsimplyconnected.) Proposition2.4. LetLdenotesomelevelcurveoffinG,orsomecomponentof@G,andletr02[0,1]bethevaluethatjfjtakesonL.LetKsc(G)denotesomeclosedsetcontainedinasinglecomponentofLc.Ifz2GissufcientlyclosetoLandinthesamecomponentofLcasK,thefollowingholds. jf(z)j6=r0. zisanon-criticallevelcurveoffinG. KandLareindifferentfacesofz. IfKisintheunboundedfaceofL,thenKisintheunboundedfaceofz.Other-wiseKisintheboundedfaceofz. Proof. LetDdenotetheintersectionofthefaceofLwhichcontainsKwithsc(G). Case2.0.3. KiscontainedinaboundedfaceofL.Theninthiscase,DequalstheboundedfaceofLwhichcontainsK.LetK1Dbethesetofallpointsz2Dsuchthatzsatisesatleastoneofthefollowing. z2K. z2Gandf(z)=0orf(z)=1. z2Gandjf(z)j=r0. z2Gandf0(z)=0. z2Gc. 18

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K1isclosed(asitisaniteunionofclosedsets)andbounded,andthuscompact.ThereforewemayndsomecompactconnectedsetfK1DwhichcontainsK1(asimplepointsettopologyargument,relyingonthefactthatopenconnectedsetsinCarepathconnected.).WecanthenndacompactconnectedsetK2DsuchthatfK1iscontainedintheinteriorofK2(forexampleaniteunionofclosedballscontainedinD).SincefK1doesnotintersect@K2,jfjdoesnottakethevaluer0on@K2.Thussince@K2iscompact,ifweset:=infz2@K2(jjf(z)j)]TJ /F3 11.955 Tf 17.54 0 Td[(r0j),then>0.Wewillnowusethistodeterminehowclosethesufcientlyclosefromthestatementofthepropositionis.SinceLandK2arecompact,some>0maybefoundsmallenoughsothatifz2CiswithinofL,thenz2GnK2,andjjf(z)j)]TJ /F3 11.955 Tf 17.93 0 Td[(r0j2(0,). Claim2.0.1. Ifz2DislessthanawayfromL,thenzisanon-criticallevelcurveoffcontainedinG,suchthatLisintheunboundedfaceofz,andK2isintheboundedfaceofz,andjf(z)j6=r0.Letz2DbelessthanawayfromL.Bythedenitionof,jjf(z)j)]TJ /F3 11.955 Tf 17.95 0 Td[(r0j2(0,)(andthusjf(z)j6=r0),sobydenitionof,zdoesnotintersect@K2.SincezdoesintersectK2c(namelyatz),andzisconnected,wemayconcludethatzdoesnotintersectK2.AndK2isconnected,soK2isentirelycontainedinoneofthefacesofz.SinceK2containsallcriticalpointsoffinD,zisanon-criticallevelcurveoff,andthereforehasonlyoneboundedface.LetFdenotetheboundedfaceofz.BytheMaximumModulusTheoremFmustcontaineitherazeroorpoleoff,orapointinGc.ButeachzeroandpoleoffandpointinGcwhichisinDiscontainedinK2,sowemayconcludethatK2iscontainedinF.AndsinceziscontainedinD,andDisaboundedfaceofL,Liscontainedintheunboundedfaceofz. Case2.0.4. KiscontainedintheunboundedfaceofL.Weuseasimilarargumentaswiththepreviouscase,butwewanttoconstructK2inawaythatwillguaranteethatwhenwehavefoundour,ifz2DiswithinofL,then 19

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K2willbecontainedintheunboundedfaceofz.ThereforewedeneK1identicallyasbeforeexceptinadditionweinclude1inK1.ConstructingK2asbefore,weobtainaclosedconnectedsetcontainedinDwhichcontainsK1initsinterior.Letandbedenedasabove.Thesameargumentasabovewithminorchangesallowsustoconcludeifz2DiswithinofL,thenzisanon-criticallevelcurveoffinGwithjfj6=r0onz,andLisintheboundedfaceofz,andK2(andthereforeK)isintheunboundedfaceofz. ThenexttheoremstatesthatifanytwolevelcurvesoffinGareexteriortoeachother,thenthereisacriticallevelcurveoffinGwhichcontainsthetwolevelcurvesindifferentboundedfaces.WewillconcludefromthisthatifweremoveallcriticallevelcurvesfromG,eachcomponentoftheremainingsetwillbeconformallyequivalenttoadiskoranannulus.If,inaddition,weremovethezerosandpolesofffromG,theneachcomponentoftheremainingsetwillbeconformallyequivalenttoanannulus. Denition7. If1and2arelevelcurvesoffinGthenwesay12if1liesinoneoftheboundedfacesof2.LetDbeanopensub-setofG.If1isacriticallevelcurveoffcontainedinD,thenwesaythat1is-maximalwithrespecttoDifthereisnoothercriticallevelcurve2offcontainedinDsuchthat12. Theorem2.1. LeteachofL1andL2bealevelcurveoffcontainedinGoracomponentof@GwhichisadjacenttoaboundedcomponentofGc.LetF1denotetheunboundedfaceofL1andF2theunboundedfaceofL2.IfL1F2,andL2F1,thenthereissomew2GwhichliesinF1\F2,suchthatf0(w)=0andL1andL2arecontainedindifferentboundedfacesofw. Proof. Dene1:=jf(L1)j,and2:=jf(L2)j. Case2.1.1. BothL1andL2arelevelcurvesoffinG.G\F1\F2isopen,andbyLemma 1 ,itisconnected.Fromthisitiseasytoshowthatwecanndapath:[0,1]!Gsuchthat(0)2L1and(1)2L2,andforall 20

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r2(0,1),(r)2G\F1\F2.DeneA(0,1)bethesetsuchthatr2Aifandonlyif(r)containsL1inoneofitsboundedfacesandL2initsunboundedface.ClearlyifLisanylevelcurveoffinGsuchthatL1andL2areindifferentfacesofL,thenLintersectsthepath.ThusProposition 2.4 guaranteesthatAisnon-empty.Soifwedener1:=sup(r2(0,1):r2A),wehaver12(0,1]. Claim2.1.1. r1<1.Proposition 2.4 alsoimpliesthatwemayndsomes2(0,1)suchthat(s)containsL2inoneofitsboundedfacesandL1initsunboundedface.LetDdenotethefaceof(s)whichcontainsL2.Since(s)andL2arecompact,thedistancebetween(s)andL2isgreaterthanzero,so,becauseiscontinuous,thereissome>0suchthatifr2(1)]TJ /F6 11.955 Tf 13 0 Td[(,1],(r)iscontainedinD.Thereforeforallr2(1)]TJ /F6 11.955 Tf 13 0 Td[(,1),(r)iscontainedinD,sotheboundedfacesof(r)arecontainedinD.Thusforallr2(1)]TJ /F6 11.955 Tf 11.95 0 Td[(,1),risnotinA.Fromthiswemayconcludethatr1<1.Wewillnowshowthat(r1)containsacriticalpointoff,andthatL1andL2arecontainedindifferentboundedfacesof(r1). Claim2.1.2. L1(r1).Since(r1)2F1\F2,(r1)F1\F2.SupposebywayofcontradictionthatL1iscontainedintheunboundedfaceof(r1).ThenbyProposition 2.4 ,thereissomenon-criticallevelcurve1offcontainedinGsuchthat(r1)iscontainedintheboundedfaceof1andL1iscontainedintheunboundedfaceof1.Thenonemayeasilyusethecontinuityoftoshowthereissomes2(0,r1)suchthatforeachr2(s,r1],(r)doesnotcontainL1inanyofitsboundedfaces(since(r)isintheboundedfaceof1).Thereforeforeachr2(s,r1],r=2A,whichisacontradictionofthedenitionofr1.ThusweconcludethatL1(r1). Claim2.1.3. L2(r1). 21

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SupposebywayofcontradictionthatL2isintheunboundedfaceof(r1).ThenProposition 2.4 givesthatthereissomenon-criticallevelcurve2offinGsuchthat(r1)intheboundedfaceof2,andL2intheunboundedfaceof2.Thus(r1)isintheboundedfaceof2,and(1)isintheunboundedfaceof2,sobythecontinuityof,thereissomer2(r1,1)suchthat(r)22(andthus(r)=2).ButL1isinoneoftheboundedfacesof(r1),soL1isintheboundedfaceof2aswell,andthusr2A.Howeverthisisacontradictionofthedenitionofr1.WeconcludethatL2iscontainedinoneoftheboundedfacesof(r1).Thatis,L2(r1).ThusL1andL2areeachcontainedinboundedfacesof(r1). Claim2.1.4. L1andL2arecontainedindifferentboundedfacesof(r1).WeuseanalmostidenticalargumenttothatfoundinClaim 2.1.2 .SupposebywayofcontradictionthatL1andL2arecontainedinthesameboundedfaceof(r1).LetDdenotethefaceof(r1)containingL1andL2.AgainbyProposition 2.4 ,thereissomenon-criticallevelcurve3offcontainedinDsuchthatL1andL2areintheboundedfaceof3,and(r1)isintheunboundedfaceof3.Therefore(0)isintheboundedfaceof3and(r1)isintheunboundedfaceof3.Thusbythecontinuityofthereissomes2(0,r1)suchthatforeachr2(s,r1],(r)isintheunboundedfaceof3.Fixforthemomentsomer2(s,r1].(r)iscontainedintheunboundedfaceof3,andL1andL2arebothcontainedintheboundedfaceof3.Therefore,since(r)doesnotintersect3,L1isintheboundedfaceof(r)ifandonlyifL2isaswell.Thusr=2Aforeachr2(s,r1],whichcontradictsthedenitionofr1.WeconcludethatL1andL2arecontainedindifferentboundedfaces(r1),whichimpliesthat(r1)containsazerooff0byProposition 2.1 ,andwearedone. Case2.1.2. AtleastoneofL1andL2arecomponentsof@G.SupposethatL1isacomponentof@G.L2iscompact,sobyProposition 2.4 ,wemayndanon-criticallevelcurveL10offinGsuchthatL1iscontainedinthebounded 22

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faceofL10andL2iscontainedintheunboundedfaceofL10.IfL2isacomponentof@G,thesameuseofProposition 2.4 givesusalevelcurveL20offinGsuchthatL2iscontainedintheboundedfaceofL20andL10(andthusL1)iscontainedintheunboundedfaceofL10.ThenwecanusetheprecedingcasewithL10andL20,whichthengivesusthedesiredresultforL1andL2. Letusnowextendthe-orderingtotheboundedcomponentsof@Gasfollows. Denition8. IfKGisacomponentof@G,andLisalevelcurveoffinGsuchthatKiscontainedinoneoftheboundedfacesofL,thenwewriteKL.Nextwehavetwocorollariestotheprevioustheoremguaranteeingtheexistenceofaunique-maximalcriticallevelcurveoffincertainregionsofG,butrstadenition. Denition9. DeneB:=fw2G:f0(w)=0orf(w)=0orf(w)=1g,anddeneC:= [w2Bw![fw2@G:wisin@DforsomeboundedcomponentDofGcg. Corollary2. Thereisaunique-maximalcomponentofC. Proof. BytheMaximumModulusTheorem,Cisnon-empty.SinceChasnitelymanycomponents,itmusthaveatleastone-maximalcomponent.SupposebywayofcontradictionthattherearetwodistinctcomponentsL1andL2ofCwhichareboth-maximal.Theneachisintheunboundedcomponentoftheother,soTheorem 2.1 guaranteesthatthereissomeothercriticallevelcurve0CsuchthatLi0fori=1,2.Butthisisacontradictionofthe-maximalityofL1andL2. Moregenerallywehavethefollowingcorollary. 23

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Corollary3. IfDisaboundedfaceofthenthereisauniquecomponentofCinDwhichis-maximalwithrespecttoD. Proof. JustreplaceGwithD,andthenthedesiredresultisjusttheresultfromCorollary 2 inthenewsetting. Inlightofourfollowingnalresult,thenamecriticallevelcurvehasnewmeaning.ThefollowingtheoremsaysthatonanycomponentofGnC,fisverysimple,beingconformallyequivalenttothefunctionz7!znforsomen2f1,2,...g.Firstabitofnotation. Denition10. Foranypath,let)]TJ /F6 11.955 Tf 9.3 0 Td[(denotethesamecurveequippedwiththeoppositeorientation. Theorem2.2. LetDbeacomponentofGnC.Thenthefollowinghold. 1. DisconformallyequivalenttosomeannulusA. 2. LetE1denotetheinnerboundaryofD,andletE2denotetheouterboundaryofD.Thenthereissome1,22[0,1]suchthat16=2,andjfj1onE1,andjfj2onE2. 3. Leti1,i22f1,2gbechosensothati1
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boundaryofDandE2denotetheexteriorboundaryofD.EachcomponentoftheboundaryofDiscontainedinalevelcurveofforacomponentof@G.Thereforewemaydene12[0,1]tobethevalueofjfjonE1and22[0,1]tobethevalueofjfjonE2.BytheMaximumModulusTheoremsinceDdoesnotcontainazeroorpoleoffandDG,weconcludethat16=2.Assumethroughoutthat1<2,otherwisemaketheappropriateminorchanges. Claim2.2.1. ThereissomeN2f1,2,...gsuchthatforanyz2D,thechangeinarg(f)aszistraversedinthepositivedirectionisexactly2N.Letz1,z22Dbegivensuchthatz16=z1.SinceDcontainsnocriticalpointsoff,z1andz2arenon-criticallevelcurvesoff,andTheorem 2.1 impliesthateitherz1z2orz2z1.Renamez1andz2ifnecessarysothatz1z2.BytheMaximumModulusTheorem,theboundedfaceofz1containssomepointinC.HoweverDcontainsnopointsofC,soE1isacontainedintheboundedfaceofz1.LetFigure 2-4 (attheendofthischapter)representthischoiceofD,z1,andz2.LetdenotethepathinFigure 2-5 .SincemayberetractedtoapointinD,andDcontainsnozeroorpoleoff,thetotalchangeinarg(f)alongiszero.Thereforethesumofthechangesinarg(f)alongz2and)]TJ /F4 11.955 Tf 9.3 0 Td[(z1equalszero.Thereforethechangeinarg(f)alongz1isthesameasthechangeinarg(f)alongz2.Weconcludethatthechangeinarg(f)alongzisindependentofthechoiceofz2D.Letdenotethiscommonnumber.Itiswellknown(seeforexample[ 13 ])thatifisanyclosedpathinCandgisafunctionanalyticonwithnozeroson,thechangeinarg(g)alongiswelldenedandthereissomen2Zsuchthatthechangeinarg(g)alongis2n.ThuswemaychooseN2Zsuchthat=2N.Bythesameargumentasabove,thechangesinarg(f)alongE1andE2areboth2Naswell.FurthermoresinceDdoesnotcontainanyzerooff0,fisinjectiveateachpointinD,soifLissomelevelcurveoffinD,arg(f)iseitherstrictlyincreasingor 25

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strictlydecreasingasListraversedinthepositivedirection.ThereforeN6=0,soN2f1,2,...g.Letusnowadopttheconventionthatunlessotherwisespecied,anytimewearereferringtoalevelcurveoffinDasapath,wearetraversingthatlevelcurveinthedirectioninwhicharg(f)isincreasing.Bytheclaimabove,thisorientationiseitherthepositiveorientationforalllevelcurvesoffinDorthenegativeorientationforalllevelcurvesoffinD.Thuswiththisassumedorientationwemaysaythatthechangeinarg(f)alonganylevelcurveLinDis2NforsomeN2f1,2,...gwhichisindependentofL. Claim2.2.2. LetbeaclosedpathinD.Thenthechangeinarg(f)alongisanintegermultipleof2N.LetkdenotethenumberoftimeswindsaroundE1.ThensinceDcontainsnozerosorpolesoff,theArgumentPrincipleimpliesthatthechangeinarg(f)alongisktimesthechangeinarg(f)alongE1.Thusthechangeinarg(f)alongisk2N.Wenowwishtodenetheconformalmappingdescribedinthestatementofthetheorem.Fixsomez0inDatwhichftakesapositiverealvalue.(Toseethatsuchapointz0exists,observethatforanyz2D,thechangeinarg(f)alongzis2N,sothereareNdifferentpointsinzatwhichftakespositiverealvalues.)Wewishtodeneamap:D!Cwhichwewillshowisaconformalmapwith(D)=ann(0;11 N,21 N).Forw2D,let:[0,1]!Dbeapathsuchthat(0)=z0and(1)=w.Denetobethechangeinarg(f)along.Thendene(w):=jf(w)j1 Nei N.Thismaybeshowntobethevalueatwoftheanalyticcontinuationoff1 Nalongthepath,andthus(ifwelldened)isanalyticinaneighborhoodofw.Thereforeifdenedassuchiswelldened,thenisanalyticonD.Letw2Dbegiven,andlet1and2bepathsinDfromz0tow.Let1(w)denotethevalueofatwobtainedusing=1andlet2(w)denotethevalueofatwobtainedbyusing=2.Let1denotethechangeinarg(f)along1andlet2denotethechangeinarg(f)along2.Let3denotethe 26

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pathinDobtainedbytraversing1andthentraversing)]TJ /F6 11.955 Tf 9.3 0 Td[(2.3isaclosedpathinD,sobytheclaimabove,thechangeinarg(f)along3(whichisthechangeinarg(f)along1minusthechangeinarg(f)along2)isanintegermultipleof2N.Thus1=2+k2Nforsomek2Z.Thus1(w)=jf(w)j1 Nei1 N=jf(w)j1 Nei2+k2N N=jf(w)j1 Nei2 Nek2N N=2(w).Thereforewhetherwedene(w)using1orusing2weobtainthesamevalue.(NotethatweareessentiallyjustshowingthatwemaytakeanNthrootoffonD.) Claim2.2.3. (D)=ann(0;11 N,21 N).NotethatbytheMaximumModulusTheorem,foreachz2D,sinceDGandDdoesnotcontainanyzeroorpoleoff,jf(z)j2(1,2),andthusj(z)j2(11 N,21 N).Therefore(D)ann(0;11 N,21 N).Let2ann(0;11 N,21 N)begiven.Choosesomex2Dsuchthatjf(x)j1 N=jj.Suchanxexistsbecausejfj1onE1andjfj2onE2,andjfjiscontinuous.Let2[0,2)besuchthatarg()=arg((x))+.(Andthus(x)ei=.)Sincethechangeinarg(f)alongxis2NwithN6=0,thereissomepointx0inxsuchthatifxistraversedfromxtox0,thenthechangeinarg(f)alongthisportionofxisexactlyN.Onecaneasilythenshowthat(x0)=.Thereforeweconcludethat(D)=ann(0;11 N,21 N). Claim2.2.4. isinjective.Letw1,w22Dbegivensuchthat(w1)=(w2).Fromthedenitionof,jf(w1)j=jf(w2)j.TheMaximumModulusTheoremandTheorem 2.1 togetherimplythatthereisonlyonelevelcurveoffinDonwhichjfjtakesthevaluejf(w1)j,andthereforew1andw2areinthesamelevelcurveoff.LetLdenotethelevelcurveoffinDwhichcontainsbothw1andw2.Let1beapathinDfromz0tow1.Let2beapathobtainedbytraversingLinthedirectionofincreaseofarg(f)fromw1tow2.(Ifw1=w2 27

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let2beconstant.)Let3denotethepathfromz0tow2obtainedbyrsttraversing1andthentraversing2.Letidenotethechangeinarg(f)alongifori2f1,2g.Sincearg(f)isstrictlyincreasingas2istraversed,andthetotalchangeinarg(f)asallofListraversedis2N,weconcludethat22[0,2N).Since(w1)=(w2),wehavethefollowingequation,calculating(w1)using1,andcalculating(w2)using3.jf(w1)j1 Nei1 N=(w1)=(w2)=jf(w2)j1 Nei1+2 N=jf(w1)j1 Nei1 Nei2 N.Dividingbothsidesbyjf(w1)j1 Nei1 N,weobtain1=ei2 N.Thus2isanintegermultipleof2N.Since22[0,2N),weconcludethat2=0,andthusw1=w2.ItremainstoshowthatextendscontinuouslyallpointsintheboundaryofDexceptpossiblythezerosoff0inE1.Toseethiswejustobservethatthedenitionofmaybedenedinthesamewayasaboveforallpointsin@D.Thisextensioniswelldened(bythesameargumentusedabove)exceptatthezerosoff0inE1.Ifzisoneofthezerosoff0inE1,thenforanypath:[0,1]!Gsuchthat([0,1))D,and(1)=z,itmaybeshownusingthecontinuityoffthatlimr!1)]TJ /F6 11.955 Tf 8.24 5.81 Td[(((r))existsbythecompactnessofcl(ann(0;i11 N,i21 N)). 28

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Figure2-1. AdmissibleGraphs Figure2-2. Non-AdmissibleGraphs Figure2-3. f(z)=z5)]TJ /F4 11.955 Tf 11.95 0 Td[(1 29

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Figure2-4. D,z1,andz2 Figure2-5. Denitionof 30

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CHAPTER3THEPOSSIBLELEVELCURVECONFIGURATIONSOFAMEROMORPHICFUNCTIONWewillnowconsideronlymeromorphicfunctionswhosedomainsaresimplyconnected. Denition11. LetGbeanopenboundedsimplyconnectedsetinC,andletf:G!Cbemeromorphic,andsuchthatfcanbeextendedtoanmeromorphicfunctiononanopensetcontainingtheclosureofG.Callsuchapair(f,G)afunctionelement. Say(f,G)isaspecialtypefunctionelementifjfj1andf06=0on@G. If(f1,G1)and(f2,G2)arefunctionelements,andthereissomeconformalmap:G1!G2suchthatf1f2,thenwesaythat(f1,G1)and(f2,G2)areconformallyequivalent,andwewrite(f1,G1)(f2,G2).Itiseasytoseethatisanequivalencerelationonthecollectionofallfunctionelements,andwemakethefollowingdenition. Denition12. LetH0denotethesetofallspecialtypefunctionelements,anddeneH:=H0=.LetH0aH0denotethesetofallspecialtypefunctionelements(f,G)suchthatfisanalyticonG,anddeneHa:=H0a=.InChapter 4 ,wewillshowthattwospecialtypefunctionelementsareinthesamememberofHifandonlyiftheyhavethesamelevelcurvestructure.Wewillseethattofullydescribethecongurationoflevelcurvesofaspecialtypefunctionelement(f,G),itsufcestoconsideronlythecongurationofthecriticallevelcurvesoff.Inordertorigorouslydenethecongurationofcriticallevelcurvesof(f,G),inthenextsectionwewilldeneamathematicalobjectPC(forPossibleLevelCurveCongurations)whichwillparameterizethedifferentpossiblelevelcurvecongurationsofaspecialtypefunctionelement. 3.1ConstructionofPCWebeginbydeningasetPwhichwillrepresentthedifferentpossiblegraphsonemayobtainasalevelcurveofaspecialtypefunctionelement.MembersofPare 31

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certainconnectednitegraphs(thatis,graphswithnitelymanyverticeswithnitelymanyedges),andmaybeviewedassub-setsofC,butaredenedmoduloorientationpreservinghomeomorphism(whichwillbedenedshortly).WewillnowdescribewhichnitegraphsarecontainedinP.Werstincludenitegraphsconsistingofasinglevertexandnoedges.Ofcourseallsinglepointsarethesamemodulohomeomorphism,sothereisonlyonememberofPwhichconsistsofasinglepoint.Beyondthis,aconnectednitegraphembeddedinCiscontainedinPifandonlyifithasthefollowingproperties. Eachedgeofisincidenttoatleastoneboundedfaceof. Foreachvertexvof,thenumberofedgesofincidenttovisevenandgreaterthan2(wherewecountanedgetwiceifbothendpointsoftheedgeareatv).WhenmoddingoutourgraphsembeddedinCbyorientationpreservinghomeomorphisms,wemeanthattwographswhichmeettheaboverestrictionsareconsideredthesameifthereisanorientationpreservinghomeomorphismofCtoitselfwhichmapstheonegraphtotheother.(Inthesettingofaspecialtypelevelcurve(f,G),thesinglepointswillbeusedtorepresentzerosorpolesoff,andthegraphswillbeusedtorepresentthecriticallevelcurvesoff.) Note: ThroughoutwhenwerefertothesinglepointelementofP,orconstructionsfromthatelement,wewilljustrefertoitasw,thoughfwgmaybetechnicallymoreaccurate.GiventhatthemembersofPwillbeusedtohelprepresentthecriticallevelcurvesofaspecialtypefunctionelement(f,G),wenowformanothersetPbyassociatingsomeauxiliarydatatothemembersofP.ToeachmemberofP,wewillassociateauxiliarydatatorepresentthefollowing. Themodulusoffonthelevelcurvebeingrepresented. Thepointsinthelevelcurvebeingrepresentedatwhichftakesnon-negativerealvalues.(Thesepointswewillcalldistinguishedpointsofthegraph.)IfthememberofPinquestionisthesinglepointmember,itwillbeusedtorepresenta 32

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zeroorpoleoff,andisthusautomaticallydistinguished.(Note:wewillview1asanon-negativerealvalue.)TothemembersofPwhicharenotsinglepoints,wewilladditionallyassociateauxiliarydatatorepresentthefollowing. Thenumberofzerosminusthenumberofpolesineachboundedfaceofthelevelcurvebeingrepresented.(Thiswillofcoursebeequaltothenumberofdistinguishedpointsintheboundaryofthatface.) Theargumentoffateachvertex(criticalpointoff)ofthelevelcurvebeingrepresented.WebeginthisprocesswiththesinglepointmembersofP.LetwdenotethesinglepointmemberofP.Fromw,wewillconstructamemberhwiPofP.Wedothisbyassociatingthefollowingpiecesofdatatow. WedeneH(hwiP)tobeavalueinf0,1g(dependingonwhetherhwiPwillrepresentazeroofforapoleoff). WewriteZ(hwiP)=nforsomenon-zeron2Z.Thisrepresentsthemultiplicityofthepointbeingrepresentedasazeroorpoleoff(positiveifazero,negativeifapole). Wesaythatwisdistinguishedwithmultiplicityjnjtorepresentthatfisnon-negativerealonw,andtheramicationoffatwisjnj.TheresultingobjectwedenotehwiP.IfisamemberofPwhichisnotthesinglepoint,thenweconstructamemberhiPofPfrombyassociatingthefollowingpiecesofdatato. WedeneH(hiP)=forsomevalue2(0,1)todenotethevalueofjfjon. IfDisaboundedfaceof,weassociateanintegerz(D)2Znf0g.(ThisrepresentsthenumberofzerosoffinDminusthenumberofpolesoffinD.)IfD1,D2,...,Dkdenotetheboundedfaceof,wedeneZ(hiP)=kXi=1z(Di).ThisassignmentmustbedoneinsuchawaythatZ(hiP)6=0andifD1andD2areboundedfacesofwhichshareacommonedge,thenz(D1)andz(D2)arenotbothpositiveorbothnegative.(ThisisthecaseforlevelcurvesoffinviewoftheOpenMappingTheorem.) 33

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ForeachboundedfaceDof,wedistinguishz(D)points(butnitelymany)in@D(torepresentthepointsinatwhichftakesnon-negativerealvalues). Ifx2isavertexof,wedesignateavaluea(x)2[0,2).(Thiswillrepresenttheargumentoffatx.)Werequirethatthisassignmentfollowsthefollowingrules. Foravertexxof,a(x)=0ifandonlyifxisadistinguishedpoint. IfDisafaceof,andz(D)>0,andx1,x2areverticesofin@Dsuchthata(x1)a(x2),thenthereissomedistinguishedpointz2@Dsuchthatx1,z,x2iswritteninincreasingorderastheyappearin@D.(Thisreectsthefactthatifisalevelcurveoff,andDcontainsmorezerosoffthanpolesoff,thentheargumentoffisincreasingas@Distraversedwithpositiveorientation.) IfDisafaceof,andz(D)<0,andx1,x2areverticesofin@Dsuchthata(x1)a(x2),thenthereissomedistinguishedpointz2@Dsuchthatx2,z,x1iswritteninincreasingorderastheyappearin@D.(Thisreectsthefactthatifisalevelcurveoff,andDcontainsmorepolesoffthanzerosoff,thentheargumentoffisdecreasingas@Distraversedwithpositiveorientation.)TheresultingobjectwiththeaboveauxiliarydatawedenotehiP,andwedenePtobethesetofallsuchhiPandhwiP.WealsodenePaPbyhwiP2PaifandonlyifZ(hwiP)>0,andhiP2Paifandonlyifz(D)>0foreachboundedfaceDof.Throughoutthispaper,hwiPwillbeusedtorefertosinglepointmembersofP,hiPwillbeusedforgraphmembersofP,andhiPwillbeusedwhenwedonotwishtodistinguishbetweenthetwotypesofmembersofP.WewillnowconstructPC.EachmemberofPCwillbeacollectionofmembersofParrangedindifferentwaysaccordingtocertainrules,withcertainauxiliarydatawhichwillbedescribed.Asnotedbefore,thiswillbeusedtorepresentthedifferentwaysinwhichthecriticallevelcurvesofaspecialtypefunctionelementmayliewithrespecttoeachother.Therearetwostepstothis.First,determinewhichgraphslieinwhichboundedfacesofwhichothergraphs,andsecond,determinetheorientationsofeachgraphwithrespecttotheothers.Webeginbydescribingthedifferentwaysinwhichthegraphsmayliewithrespecttoeachotherrecursively. Note: EachmemberofhiPofPwillgiverisetopossiblymultipledifferentmembersofPC,butwhenitwillnotcauseconfusion,wewillusehiPCtodenoteamemberofPC 34

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whicharisesfromhiP.AllmembersofPCwillarisefromsomememberofP,soifweusehiPCtodenotesomememberofPC,thenbyhiPwemeanthememberofPwhichgaverisetohiPC.Alevel0memberofPCwillbeasinglepointmemberofPviewedasamemberofPC,withnoadditionaldata.Forn>0,levelnmembersofPCareconstructedbytakinghiPagraphmemberofP,andassigningtoeachboundedfaceDofalevelkmemberhDiPCofPCforsomek0,thenH(hDiP)H(hiP).Furthermore,letDbeanyboundedfaceof.ThenhiPCalsocomesequippedwithasurjectivemapgDfromthedistinguishedpointsin@DtothedistinguishedpointsinD.Thismapwecallthegradientmap,(sinceinthestudyofaspecialtypefunctionelement(f,G),thegradientmapswillbedeterminedbythegradientlinesoff)andwerequirethatgDpreservetheorientationofthedistinguishedpoints.Thatis,ifDisagraphembeddedinC,andwereadthedistinguishedpointsinDoffastheyappeararoundDwhenDistraversedonefulltimeinpositiveorderfromtheoutside,letx1,...,xz(D)betheirenumerationastheyappearinthisway(ifsomevertexofDisdistinguished,thenitwillappearmorethanonetimeinthislist).Thentheremustbesomepointy12@Dwhichisdistinguishedasapointin,andwhenthedistinguishedpointsin@Darelistedbytheirappearenceinpositiveorderstartingwithy1,namelyy1,...,yz(D),thengD(yi)=xiforeachi2f1,...,z(D)g.IfDisjustasinglepointwithassociatednumber0,thengDisjustthemapthattakeseverydistinguishedpointin@Dtothatsinglepoint.(Inthecontextoflevelcurvesofmeromorphicfunctions,gD(w)=zmeansthatwandzareconnectedbyagradientlineoff.Thiskeepstrackoftheorientationofacriticallevelcurveoffwithrespecttotheothercriticallevelcurvesoff.) 35

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Note: IfamemberofPChasbeenformedatlevelnforsomen0,wedonotformitagainatanylaterlevel,sowemaysayinawelldenedwaythatamemberhasagivenlevel.WecallthisassignmentofmembersofPCtotheboundedfacesof,alongwiththeassociatedgradientmaps,hiPC.ThesetofallsuchhiPCandhwiPCwedenotePC,andwecallthisthesetofpossiblelevelcurvecongurations.WedenePCaPCtobethecollectionofmembersofPCwhichisconstructedentirelyusingmembersofPa.Thatis,hiPC2PCaifandonlyifhiP2Pa,andeachmemberofPCwhichisassignedtoaboundedfaceofisinPCa.Weadoptthesameconventionofw,orformembersofPCaswedidformembersofP,namelythatlevel0membersofPCwedenotebyhwiPC.Leveln>0membersofPCwedenotebyhiPC,andifwedonotwishtospecifythelevelofamemberofPCwewilldenoteitbyhiPC. 3.2ConstructionofWenowmakeexplicitthewayinwhichPCparameterizesthepossiblelevelcurvecongurationsofaspecialtypefunctionelement.Wedothisbydeningafunction:H0!PC.InChapter 4 ,wewillshowthattakesthesamevalueonconformallyequivalentmembersofH0,andthereforewemayviewasactingonH.ItisfairlyeasytoshowthatactingonHisinjective,andthat(Ha)PCa.Oneofthemajorgoalsofthispaperwillbetoshowthat:Ha!PCaisabijection.WenowdenetheactionofonH0.Let(f,G)besomememberofH0.RecallthatB=fw2G:f0(w)=0orf(w)=0orf(w)=1g.Proposition 2.1 showsthatforeachw2B,wiseitherasinglepoint(ifwisazerooff),orisaplanargraphofthetypeusedtoformmembersofP.WebeginbypickingmembersofPtorepresentwforeachw2B.Ifwiseitherazeroorapoleoff,thenw=fwg,andwedeneZ(hwiP):=kwherekisthemultiplicityofwasazerooff,(k<0ifwisapoleoff,)anddeneH(hwiP):=0ifw 36

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isazerooffandH(hwiP)=1ifwisapoleoff.TheresultingobjecthwiPisnowamemberofP.Ifwisnotasinglepoint,thenletDbeaboundedfaceofw.Wedenez(D):=k,wherekisthenumberofzerosoffwhicharecontainedinDminusthenumberofpolesoffinD.Wedistinguishthepointsin@Datwhichftakespositiverealvalues(therewillbeexactlyjkjofthem).Andtoeachvertexw02w,weassociatethenumbera(w0)2[0,2)wherea(w0)isthechoiceoftheargumentoff(w0)whichliesin[0,2).Clearlya(w0)=0ifandonlyifw0isdistinguished.Itisalsothecasethatifk>0,andz,z0areverticesofwwithz6=z0and0
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Case3.0.1. ThereisasingledistinctzeroorpoleoffcontainedinD,andthereisnozerooff0containedinDwhichisnotazerooff.Inthiscase,letw0denotethissingledistinctzeroorpoleoff.Z(hw0iP)=z(D),sincew0istheonlyzeroorpoleoffinD,andifk>0,thenH(hw0iP)=0H(hwiP),sowemayassociatehw0iPCtoD.FinallywedenegDtomapallthedistinguishedpointsin@Dtow0. Case3.0.2. Thereissomecriticalpointw0offinGwhichisnotazerooff,andsuchthateachmemberofBwhichisinDiseitherinw0orinoneoftheboundedfacesofw0.Proceedrecursively.Assumethathw0iPChasbeenalreadybeenformed.Sinceeachzeroandpoleoffisinsomeboundedfaceofw0,Z(hw0iPC)=z(D).Furthermore,ifk>0,thenthevalueofjfjonw0isstrictlylessthanthevalueofjfjonw,andthereforeH(hw0iP)H(hwiP),sowemayassociatehw0iPCtoD.NowwewishtodenegD.Letz2@Dbedistinguished(thusf(z)>0).ThefactthatnogradientlinesmayintersectintheregioninDwhichisexteriortow0(sincetherearenozerosoff0inthatregion)givesusthatthereisonlyasingledistinguishedpointinw0whichisconnectedtozbyaportionofagradientlineoffwhichliesentirelyinDbutexteriortow0.Callthatdistinguishedpointz02w0.ThenwedenegD(z):=z0.SincethegradientlinesoffdonotcrossintheregionofDexteriortow0,themapgDsodenedrespectstheorderofthedistinguishedpointsastheyappearin@D.Dothisassignmentprocess,anddenitionofthegradientmap,foreachboundedfaceofw.TheresultingobjectwecallhwiPC.SinceBhasnitelymanymembers,thisprocessterminates.Corollary 3 impliesthatthereissomepointw2Bsuchthateachw02Biseitherinworinoneoftheboundedfacesofw.Furthermore,ifw,w0areany 38

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twosuchpoints,itiseasytoseethathwiPCandhw0iPCasdenedaboveareequal.Thereforewemaydene(f,G):=hwiPC.ThusweclassifythewaysinwhichthecriticallevelcurvesofafunctionfmaybeconguredinitsdomainGby(f,G). 39

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CHAPTER4RESPECTSCONFORMALEQUIVALENCEOurgoalinthischapteristoshowthatconformalequivalenceofspecialtypefunctionelementsmaybedeterminedentirelybytheirrespectivelevelcurvestructures.Thatis,wehavethefollowingtheorem. Theorem4.1. If(f1,G1)and(f2,G2)aretwospecialtypefunctionelements,then(f1,G1)(f2,G2)ifandonlyif(f1,G1)=(f2,G2). Proof. Theforwardimplication(thatif(f1,G1)(f2,G2),then(f1,G1)=(f2,G2))followsfairlystraightforwardlyfromthedenitionofandthedenitionof.Let:G1!G2beaconformalmapintertwiningf1andf2(ie.f1=f2).Thenifisalevelcurveoff1inG1,then()isalevelcurveoff2inG2.Ifwisazeroorpoleorcriticalpointoff1inG1,then(w)isazeroorpoleorcriticalpointrespectivelyoff2inG2withthesamemultiplicity,andcarriesgradientlinesoff1togradientlinesoff2.Itfollowsimmediatelythattheconstructionof(f1,G1)proceedsinexactlythesamemannerastheconstructionof(f2,G2),soweproceedtothemoredifcultbackwardimplication.Assumethat(f1,G1)=(f2,G2).LethiPCdenotethismemberofPC.Fori2f1,2g,deneBi:=fw2Gi:f10(w)=0orfi(w)=0orfi(w)=1g.DeneCiGibyCi:=[w2Biw.LethiPbesomememberofPusedintheconstructionofhiPC.Fori2f1,2g,letidenotethelevelcurveoffiwhichgivesriseto.Since1and2arethesamewhenviewedasmembersofP,thereisanorientationpreservinghomeomorphism:1!2.Furthermore,ifE1issomeedgein1,andE2isthecorrespondingedgein2,thenE1andE2containthesamenumberofdistinguishedpoints.Thereforebyreparameterizing,wemayassumethatmapsthedistinguishedpointsof1tothedistinguishedpointsof2.SincewemayformthisorientationpreservinghomeomorphismforeachhiPusedtoconstructhiPC,wemaystitchthesehomeomorphismstogethertoobtainanorientationpreservinghomeomorphism:C1!C2. 40

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Furthermore,sincepreservesinformationabouthowdistinguishedpointsareconnectedbygradientlines,wemayassumethatrespectsgradientlines.Thatis,ifD1isaboundedfaceof1,andD2isthecorrespondingboundedfaceof2(thatis,D2=(D1)),andw1isadistinguishedpointin@D1,thenmaybechosensothat(gD1(w1))=gD2((w1)).Wenowshowthatwecanassumethatf1=f2onC1.Let1besomecomponentofC1.If1isjustasinglepoint,then(1)isasinglepoint,andbecause1and(1)giverisetothesamememberofP,1and(1)areeitherbothzerosorbothpolesoff1andf2respectively.Thereforef1=f2on1.Nowassumethat1isacriticallevelcurveoff1suchthatH(h1iP)2(0,1)(whereh1iPreferstothememberofPusedtoconstruct(f1,G1)whicharisesfrom1).ThenletDbeaboundedfaceof1.LetCdenotethecollectionofdistinguishedpointsandverticesof1whicharecontainedin@D.Thatis,Cisthecollectionofpointsin@Datwhichf1takespositiverealvalues,orf10=0.Thensinceisahomeomorphismwhichpreservestheinformationrecordedby,(iewhatpointsfitakespositiverealvaluesat,andwhatargumentfitakesattheverticesofthelevelcurveinquestion,andwhatthemodulusoffiisonthatlevelcurvefori2f1,2g),wehavethatf1=f2ateachpointinC.LetTbeasegmentof@DwhoseendpointsareinCandsuchthatnootherpointinTisinC.SoTdoesnotcontainanyvertexofS,exceptpossiblyattheendpoints.Letw,w0betheendpointsofTchosensothatw,T,w0iswritteninincreasingorder.Ifw0isdistinguished,viewitsargumentasbeing2ratherthan0forthemoment.Assumethatz(D)>0.Thenarg(f1())isacontinuousfunctiononTthatisincreasingasTistraversedinthepositivedirection,andthesamemaybesaidoff2on(T),anditisawellknownfactthatunderthesecircumstances,thereisahomeomorphism^mappingTto(T)suchthatarg(f1())=arg(f2(^())).Replaceby^onT.Inthiswayitmaybeseenthatf1=f2on1.Ifz(D)<0,thesameresultholdsbyananalogousarguments.Sowemayassumethatf1=f2onallofC1. 41

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WenowwishtoextendtoG1nC1insuchawaythatjG1nC1isanalytic,andf1=f2onG1nC1.WethenshowthatthisextendedisanalyticonallofG1.LetFbeacomponentofG1nC1.ByTheorem 2.2 ,Fishomeomorphictoanannulus.LetKdenotetheboundedcomponentofFc,andletndenotethenumberofzerosoff1containedinKminusthenumberofpolesoff1containedinK.LetLiandLedenotetheinteriorandexteriorboundariesofF.LetHi,He>0bethenumberssuchthatjfjHionLi,andjfjHeonLe.Assumethatn>0,andchoosesomedistinguishedpointwinLe.ThenHi0.Let(F)denotethecomponentofG2nC2whichisboundedby(Li)and(Le).Thenbythesamereasoningasabove,thereisaconformalmap2:(F)!ann(0,(Hi)1 n,(He)1 n)thatsatisesthefollowing. (2)n=f2. 2extendscontinuouslytoallpointsin(Li)and(Le)withtheexceptionofthecriticalpointsoff2in(Li). Ifzisacriticalpointoff2in(Li),and:[0,1]!G2isapathwith([0,1))(F)and(1)=z,thenlimr!12((r))exists. 2((w))>0.(Note:Wecandothisbecause(w)isadistinguishedpointin(Le).)WeextendtoFby=2)]TJ /F5 7.97 Tf 6.58 0 Td[(11.ThisisclearlyaconformalmapfromFto(F).Notethatf22)]TJ /F5 7.97 Tf 6.59 0 Td[(1(z)=znforeachpointzinann(0,(Hi)1 n,(He)1 n).Therefore,foreachz2F,f2((z))=f2(2)]TJ /F5 7.97 Tf 6.59 0 Td[(1(1(z)))=(f22)]TJ /F5 7.97 Tf 6.58 0 Td[(1)(1(z))=1(z)n=f1(z). 42

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WenowhavethatisamapfromG1toG2whichisbijective(sinceisahomeomorphismonC1andconformaloneachcomponentofG1nC1),andthatf1=f2onallofG1.WenowwishtoshowthatisanalyticonallofG1.Letz2G1nB1begiven.Ifz2G1nC1,thenwealreadyknowthatisanalyticatz.Supposethatz2C1.Sincez=2B1,zisnotacriticalpointoff,soasnotedabove,iscontinuousinaneighborhoodofz.ThenasanapplicationoftheSchwartzReectionPrinciple,wemayconcludethatisanalyticatz.ThusisanalyticatallpointsinG1exceptpossiblythenitelymanypointsinB1.LetzbeoneofthepointsinB1.Itsufcestoshowthatiscontinuousatz.Iflimw!z(w)exists,thenthislimitequals(z)becausejzisahomeomorphism.Letusassumethatthislimitdoesnotexist.Thenthereisasome>0andsomesequenceofpointsfzkg1k=1G1suchthatlimk!1zk=z,andj(zk))]TJ /F6 11.955 Tf 12.42 0 Td[((zk+1)j>foreachk.Thensincesincethereareonlynitelymanyfacesofzwhichareincidenttoz,bydroppingtoasub-sequence,wecanassumethatallzkareinthesamefacecomponentofG1nC1,orthatallzkareinz.Ifallthezkareinz,thenwehaveacontradictionbecauseisahomeomorphismonz.IfallzkareinsomesinglecomponentDofG1nC1,thenitiseasytoshowthatwemayndsomepath:[0,1]!G1suchthat([0,1))D,and(1)=z,and(1)]TJ /F5 7.97 Tf 13.51 4.71 Td[(1 k)=zkforeachk.Butbydenitionof1and2,wemayconcludethatlimr!1)]TJ /F6 11.955 Tf 8.25 5.81 Td[(((r))exists,sowehaveacontradictionandwearedone.ThusweconcludethatisanalyticonG1,andasalreadynotedisabijectionwhichintertwinesf1andf2,soweconcludethat(f1,G1)(f2,G2). Thefactobservedatthebeginningofthissectionthat(f1,G1)(f2,G2)impliesthat(f1,G1)=(f2,G2)givesusthatwemayviewasactingontheequivalenceclassesofspecialtypefunctionelements.Thatis,dening:H!PCby([(f,G)]):=(f,G)iswelldened.ThebackwardsimplicationofTheorem 4.1 (that(f1,G1)=(f2,G2)impliesthat(f1,G1)(f2,G2))givesusthat:H!PCisinjective. 43

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CHAPTER5ISSURJECTIVE:THEGENERICCASEItisfairlyeasytoshowfromthedenitionofthat(Ha)PCa.Toshowthat(Ha)=PCa,webeginbyconsideringsubsetofHawhichcontainsspecialtypefunctionelements(f,G)wherefisapolynomial. Denition13. ForGCanopensimplyconnectedset,andf:G!CanalyticonG,and>0,deneGf,:=fz2G:jf(z)j
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Denition17. Forn2aninteger,andv2Vn)]TJ /F5 7.97 Tf 6.59 0 Td[(1,letH0p,vdenotethesubsetofmembers(f,G)2H0psuchthatthecriticalvaluesoffareexactlyv(1),v(2),...,v(n)]TJ /F5 7.97 Tf 6.59 0 Td[(1).Further,deneHp,v:=H0p,v=.WenowwillworkoutanotionofcriticalvaluesforamemberofPCa.Inessence,thecriticalvaluesofamemberhiPCarethecriticalvaluesofanymemberof)]TJ /F5 7.97 Tf 6.58 0 Td[(1(hiPC).Sincewedonotyetknowthatthissetisnon-empty,wewillhavetodenethecriticalvaluesofhiPCdirectlyfromhiPC.Webeginwithsomedenitionshavingtodowithgraphs. Denition18. For2P,andwavertexin,letm(w)denotethenumberofedgesofincidenttowwherewecounttheedgetwiceifbothofitsendpointsareatw. Note: Asmentionedearlier,ifisamemberofP,andwisavertexof,thenm(w)iseven,andgreaterthanorequalto4. Denition19. ForamemberofP,andwavertexof,thenweletm(w)denotethenumberofedgesofincidenttoz.Furthermore,wesaythatwisavertexofwithmultiplicitym(w) 2)]TJ /F4 11.955 Tf 12.03 0 Td[(1.Notethatifw2isnotavertexof,thenthisdenitionstillmakessense:ifwecounttheedgeofwhichcontainswasmeetingwtwotimes(iefromeitherside),thenwewouldsaythemultiplicityofwasavertexofis2 2)]TJ /F4 11.955 Tf 11.96 0 Td[(1=0.Notethatif(f,G)isaspecialtypefunctionelement,andw2Gisazerooff0withmultiplicityk,thenfis(k+1)-to-1inaneighborhoodofw.Thereforethereare2(k+1)edgesofwwhichareincidenttow.Thusthemultiplicityofwasavertexofwis2(k+1) 2)]TJ /F4 11.955 Tf 11.96 0 Td[(1=k. Denition20. LethiPC2PCabegiven.IfhwiPCisoneofthelevel0membersofPCusedtoformhiPC,thenwesaythat0isacriticalvalueofhiPCwithmultiplicityZ(hwiP))]TJ /F4 11.955 Tf 12.48 0 Td[(1.Supposethath0iPisamemberofPusedtobuildhiPC.Thenifwisavertexof0,wesaythatH(h0iP)eia(w)isacriticalvalueofhiPCofmultiplicityequaltothemultiplicityofwasavertexof0. 45

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WiththisnotionofcriticalvaluesofamemberofPCabuiltup,wemaynowpartitionPCaasfollows. Denition21. Forv=(v(1),...,v(n)]TJ /F5 7.97 Tf 6.59 0 Td[(1))2Vn)]TJ /F5 7.97 Tf 6.59 0 Td[(1,denePCa,vtobethecollectionofmembersofPCawhosecriticalvalueslistedaccordingtomultiplicityarev(1),...,v(n)]TJ /F5 7.97 Tf 6.59 0 Td[(1),andletjPCa,vjdenotethenumberofelementsofPCa,v.(Inthecontextofpolynomials,nisthenumberofzerosofthepoynomial,andthustherewouldben)]TJ /F4 11.955 Tf 12.13 0 Td[(1criticalvaluesofthepolynomial.)FromthedenitionofcriticalvaluesofamemberofPC,itshouldbeclearthat(Hp,v)PCa,v.Thentoshowthatissurjective,weshowthat(Hp.v)=PCa,vforeachv2V.InthischapterweshowthatthisequalityholdsforanyvinadensesubsetUofVabouttobedened,andthenextendthistoallofVinChapter 6 Denition22. Fornapositiveinteger,deneUnVntobethecollectionofu=(u(1),...,u(n))2Vnsuchthat0
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twiceifbothendsmeetatthevertex.).ThuseachgraphusedwhileconstructinghiPCisthisgureeightgraph.WewilluseaninductionargumenttocountthenumberofmembersofPCa,v0.Inordertodothis,itwillbehelpfultohaveanorderingonthemembersofPusedtoconstructagivenmemberofPCa,whichwedenenow. Denition23. FixsomehiPC2PCa,andleth1iPC,...,hniPCwithn2bethemembersofPCwhichareusedinconstructinghiPC.ThenwesayiwithrespecttohiPCforeachi2f1,2,...,ng.Furthermore,ifsomehiiPChasbeenassociatedtosomeboundedfaceofsomejwhileconstructinghiPCforsomei,j,thenwesayijwithrespecttohiPC(thiswithrespecttohiPCwillusuallybesuppressedwhenthememberhiPCinquestionisclear).Weextendthistobeatransitiverelation.Thatis,ifi1i2ikforsome2kn,thenwesayi1ik.IfhiPCisamemberofPCa,v0,sinceallcriticalvaluesofhiPCarenon-zero,andthuscomefromaverticesofagraphusedinconstructinghiPC,andeachplanargraphusedintheconstructionofhiPCcontainsasinglevertex,andeachoftheseverticesgivesrisetoacriticalvalueofhiPC,theremustben)]TJ /F4 11.955 Tf 12.23 0 Td[(1distinctplanargraphsusedtoconstructhiPC.ThereforewemakethefollowingdenitioninordertobeabletorefertothevertexwhichgivesrisetoagivencriticalvalueofhiPC. Denition24. ForhiPC2PCa,v0,andi2f1,...,N)]TJ /F4 11.955 Tf 12.01 0 Td[(1g,letzidenotethepointorvertexfromwhichthecriticalvaluev0(i)arose.Furthermore,letidenotetheplanargraphwhichcontainsthevertexzi.(Notethatsincev02Un)]TJ /F5 7.97 Tf 6.58 0 Td[(1,thisiswelldened.)WenowwishtoshowthatPCa,v0hasexactly1memberifn=2,andnn)]TJ /F5 7.97 Tf 6.58 0 Td[(3distinctmembersifn3.Wewillhavetohandlethedifferentpossiblevaluesofnseparatelyupton=6.Forn6wewillbeabletomakeageneralargument. Case5.0.1. n=2.LethiPCbeamemberofPCa,v0.SincehiPChasasinglecriticalvalue,hiPCisconstructedfromasinglegureeightgraph,namely.LetDbeeitherboundedface 47

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of,andlethwiPCdenotethelevel0memberofPCassociatedwithD.Since0isnotacriticalvalueofhiPC,Z(hwiPC)=1,sothereisonlyonedistinguishedpointin@D,andthusthereisonlyonepossiblechoiceofgradientmapgD,namelytheonethattakesthesingledistinguishedpointin@Dtow.Furtherifzisthesinglevertexof,thena(z)=arg(v0(1))andH(hiP)=jv0(1)jsincev0(1)istheonlycriticalvalueofhiPC.SoallthedatapertainingtohiPCisdeterminedentirelybyv0.ThusPCa,v0containsonlyasingleelement.Forthefuturecaseswewillneedthefollowingdenition. Denition25. LethiPCbeamemberofPC,andletDdenoteoneoftheboundedfacesof.ForsomehiPusedinconstuctinghiPC,ifhiPCwereassociatedtoD,thenwesayD.Weextendthisasfollows.Ifh1iP,...,hkiPwereusedintheconstructionofhiPC,and1kD,thenwesay1D. Note: LethiPCbeanymemberofPC,leth0iPbeanymemberofPusedintheconstructionofhiPC,andletDbeanyfaceof0.AneasyinductionargumentgivesthatthenumberofsinglepointelementshwiPofPsuchthatwDisexactlyz(D)(wherethesesinglepointmembersofParecountedaccordingtomultiplicity),andthatthenumberofcriticalvaluesofhiPCwhichcomefrommembersofPCassociatedtoDisexactlyz(D))]TJ /F4 11.955 Tf 11.95 0 Td[(1. Denition26. LethiPCbeanymemberofPCa,v0.Sincev0wastakenfromUn)]TJ /F5 7.97 Tf 6.58 0 Td[(1,onlycontainsonevertexandhasonlytwoboundedfaces.LetD1denotetheboundedfaceoffromwhichfewercriticalvaluescome.LetD2denotetheotherone.Thatis,thenamingisdonesothatz(D1)z(D2).(Ifbothboundedfacesofgiverisetothesamenumberofcriticalvalues,thenthisnamingisarbitrary.) Note: ForanymemberhiPCofPCa,v0,z(D1)+z(D2)=n.Therearen)]TJ /F4 11.955 Tf 12.97 0 Td[(1totalcriticalvaluesofhiPC,andoneofthosecriticalvaluescomesfromthevertexof,son)]TJ /F4 11.955 Tf 12.72 0 Td[(2ofthemcomefromthetworegionsD1andD2.Thistogetherwiththefactthat 48

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z(D1)z(D2)immediatelygivesthatthepossiblevaluesforz(D1))]TJ /F4 11.955 Tf 12.78 0 Td[(1totakeareexactlyfk2Z:0kn)]TJ /F5 7.97 Tf 6.59 0 Td[(2 2g. Case5.0.2. n=3.Since12,andh1iPandh2iParetheonlyplanargraphmembersofPusedtoconstructh2iPC,wehavethath1iPChasbeenassociatedtoD2.LethwiPCdenotethelevelzeromemberofPCassociatedtoD1.ThenasinCase 5.0.1 ,thereisonlyonepossiblechoiceofgD1,namelytheonethatmapsthesingledistinguishedpointof@D1tow.WeclaimthatthereisonlyasinglechoiceofgD2aswell.Since1hasexactlytwoboundedfaces,andeachofthesefaceshasonlyasinglelevelzeromemberofPCassociatedtoit,wehavethatZ(h1iP)=2,andthusz(D2)=2.Andthusthereareexactlytwodistinguishedpointsin@D2.Thetwodistinguishedpointsin1areeitherbothatthevertex,orneitheratthevertex,andineithercasethereisaorientationpreservinghomeomorphismof1whichexchangesthedistinguishedpoints.Thusmoduloorientationpreservinghomeomorphismof1(andrecallthatPCisformedmodulothesehomeomorphisms)thereisonlyonepossiblechoiceofgD2.AnditiseasytoseeasinCase 5.0.1 thatalltherestofthedata(iethevaluesH()anda()take)forh2iPCisfullydeterminedbyv0(uptoorientationpreservinghomeomorphism),andthusPCa,v0hasexactlyonemember. Note: OurgeneralwayofcountingthenumberofelementsinPCa,v0whenn3willbetopartitionPCv0bythevaluez(D1))]TJ /F4 11.955 Tf 12.49 0 Td[(1takes.Foragivenvalueofz(D1))]TJ /F4 11.955 Tf 12.48 0 Td[(1,wendouthowmanywaysthez(D1))]TJ /F4 11.955 Tf 12.12 0 Td[(1differentcriticalvaluesmaybechosenfromthen)]TJ /F4 11.955 Tf 12.12 0 Td[(2criticalvaluesavailabletocomefromD1andD2(whichisofcourse)]TJ /F15 7.97 Tf 14.01 -4.37 Td[(n)]TJ /F5 7.97 Tf 6.59 0 Td[(2z(D1))]TJ /F5 7.97 Tf 6.58 0 Td[(1).ForthatchoiceofcriticalvaluescomingfromD1,wecountthenumberofmembersofPCwhichmaybeassociatedtoD1andthenumberwhichmaybeassociatedtoD2(anaturalinductionstep).WethencountthenumberofchoicesofgD1andofgD2(z(D1)andz(D2)respectively,exceptinthecasewherez(D1))]TJ /F4 11.955 Tf 12.29 0 Td[(1=1orz(D2))]TJ /F4 11.955 Tf 12.29 0 Td[(1=1aswewillsee). 49

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WethenmultiplythesenumberstondthenumberofmembersofPCv0withthegivenvalueofz(D1))]TJ /F4 11.955 Tf 11.95 0 Td[(1. Case5.0.3. n=4.Asnotedabove,theonlypossiblevaluesofz(D1))]TJ /F4 11.955 Tf 11.95 0 Td[(1inthiscaseare0and1.Ifz(D1))]TJ /F4 11.955 Tf 13 0 Td[(1=0,thenthereisasinglelevel0memberofPCwhichcouldbeassociatedtoD1.LetthismemberbecalledhwiPC.ThenZ(hwiP)=1,soz(D)=1.ThusthereisasinglechoiceofgD1,namelythemapthattakesthesingledistinguishedpointin@D1tow.Ontheotherhand,sincez(D1))]TJ /F4 11.955 Tf 12.57 0 Td[(1=0,z(D2))]TJ /F4 11.955 Tf 12.58 0 Td[(1=2,andfromCase 5.0.2 weknowthatthenthereisonlyasinglememberofPCwhosecriticalvaluesarev0(1),v0(2).Howeversincez(D2))]TJ /F4 11.955 Tf 12.4 0 Td[(1=2,@D2has3distinguishedpoints,sotherearethreedifferentchoicesofgD2.Thusthereare)]TJ /F5 7.97 Tf 5.48 -4.38 Td[(201113=3membersofPCv0forwhichz(D1))]TJ /F4 11.955 Tf 11.96 0 Td[(1=0.Supposez(D1))]TJ /F4 11.955 Tf 12.15 0 Td[(1=1,andthusz(D2))]TJ /F4 11.955 Tf 12.15 0 Td[(1=1.Hencethereare)]TJ /F5 7.97 Tf 5.48 -4.37 Td[(21=2possiblechoicesofthecriticalvaluewhichcomesfromD1.BytheworkdoneforCase 5.0.1 ,thereisasinglepossiblememberofPCwhichmaybeassociatedtoD1andasinglememberofPCwhichmaybeassociatedtoD2.AlsobytheworkdoneinCase 5.0.1 ,thereisasinglepossiblechoiceofgD1andasinglepossiblechoiceofgD2.Andhencewecount)]TJ /F5 7.97 Tf 5.48 -4.38 Td[(211111=2membersofPCa,v0forwhichz(D1))]TJ /F4 11.955 Tf 11.97 0 Td[(1=1.Butinthiscase,onecriticalvaluecomesfromD1andonecomesfromD2,sowemayswitchtherolesofD1andD2withoutbreakingtherestrictionz(D1)z(D2).Thatis,weareovercountingbyafactorof2.Sonallywegetthatthereis21 2=1memberofPCv0forwhichz(D1))]TJ /F4 11.955 Tf 11.95 0 Td[(1=1.SojPCa,v0j=3+1=4=44)]TJ /F5 7.97 Tf 6.59 0 Td[(3. Case5.0.4. n=5.Hereagaintheonlypossiblevaluesofz(D1))]TJ /F4 11.955 Tf 11.98 0 Td[(1are0and1.Forthesakeofbrevitywewillleaveoutmuchoftheexplanationthatcaneasilybetranslatedfromthepreviouscase. 50

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Ifz(D1))]TJ /F4 11.955 Tf 12.81 0 Td[(1=0,thereisonlyonememberofPCwhichmaybeassociatedtoD1,andonechoiceofgD1.FromCase 5.0.3 ,above(preguringaninductiveargumenthere),thereare4membersofPCthatmaybeassociatedtoD2,andforeachchoiceofthememberofPCassociatedtoD2,thereare4possiblechoicesofgD2.Hencethereare)]TJ /F5 7.97 Tf 5.48 -4.38 Td[(301144=16membersforwhichz(D1))]TJ /F4 11.955 Tf 11.96 0 Td[(1=0.Ifz(D1))]TJ /F4 11.955 Tf 12.76 0 Td[(1=1,thereare)]TJ /F5 7.97 Tf 5.48 -4.38 Td[(31differentchoicesofthecriticalvaluethatcomesfromD1.Forthatchoice,thereisasinglememberofPCawhichmaybeassociatedtoD1,andasinglechoiceofgD1.ForthatchoiceofthecriticalvalueinD1(andthusthetwocriticalvalueswhichcomefromD2),thereisasinglememberofPCawhichmaybeassociatedtoD2,and3choicesofgD2.HencePCa,v0has)]TJ /F5 7.97 Tf 5.48 -4.38 Td[(311113=9membersforwhichz(D1))]TJ /F4 11.955 Tf 11.96 0 Td[(1=1.SojPCa,v0j=16+9=25=55)]TJ /F5 7.97 Tf 6.59 0 Td[(3. Case5.0.5. n6.Inthefollowingcalculations,therstnumberwillbethenumberofwaysofchoosingthecriticalvalueswhichcomefromD1.ThesecondnumberwillbethenumberofmembersofPCwhichmaybeassociatedtoD1(inductionstep)forthegivenchoiceofcriticalvaluescomingfromD1.ThethirdnumberwillbethenumberofpossiblechoicesofgD1.ThefourthnumberwillbethenumberofmembersofPCwhichmaybeassociatedtoD2(inductionstep)ThefthnumberwillbethenumberofpossiblechoicesofgD2.Assumerstthatnisodd.Thenz(D1))]TJ /F4 11.955 Tf 11.95 0 Td[(1cantakeanyvalueinthesetf0,1,...,(n)]TJ /F4 11.955 Tf 11.95 0 Td[(2))]TJ /F4 11.955 Tf 11.95 0 Td[(1 2=n)]TJ /F4 11.955 Tf 11.95 0 Td[(3 2g.ThenumberofmembersofPCa,v0forwhichz(D1))]TJ /F4 11.955 Tf 11.96 0 Td[(1=0isn)]TJ /F4 11.955 Tf 11.96 0 Td[(2011(n)]TJ /F4 11.955 Tf 11.95 0 Td[(2+1)(n)]TJ /F5 7.97 Tf 6.59 0 Td[(2+1))]TJ /F5 7.97 Tf 6.59 0 Td[(3(n)]TJ /F4 11.955 Tf 11.96 0 Td[(2+1)=(n)]TJ /F4 11.955 Tf 11.96 0 Td[(1)n)]TJ /F5 7.97 Tf 6.59 0 Td[(3.ThenumberofmembersofPCa,v0forwhichz(D1))]TJ /F4 11.955 Tf 11.96 0 Td[(1=1is 51

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n)]TJ /F4 11.955 Tf 11.96 0 Td[(2111(n)]TJ /F4 11.955 Tf 11.95 0 Td[(2)]TJ /F4 11.955 Tf 11.95 0 Td[(1+1)(n)]TJ /F5 7.97 Tf 6.59 0 Td[(2)]TJ /F5 7.97 Tf 6.59 0 Td[(1+1))]TJ /F5 7.97 Tf 6.59 0 Td[(3(n)]TJ /F4 11.955 Tf 11.95 0 Td[(2)]TJ /F4 11.955 Tf 11.95 0 Td[(1+1),whichisequalton)]TJ /F4 11.955 Tf 11.96 0 Td[(21(n)]TJ /F4 11.955 Tf 11.96 0 Td[(2)n)]TJ /F5 7.97 Tf 6.59 0 Td[(4.If2in)]TJ /F5 7.97 Tf 6.59 0 Td[(3 2,thenthenumberofmembersofPCa,v0forwhichz(D1))]TJ /F4 11.955 Tf 11.96 0 Td[(1=iisn)]TJ /F4 11.955 Tf 11.96 0 Td[(2i(i+1)i+1)]TJ /F5 7.97 Tf 6.58 0 Td[(3(i+1)(n)]TJ /F4 11.955 Tf 11.95 0 Td[(2)]TJ /F3 11.955 Tf 11.96 0 Td[(i+1)(n)]TJ /F5 7.97 Tf 6.59 0 Td[(2)]TJ /F15 7.97 Tf 6.59 0 Td[(i+1))]TJ /F5 7.97 Tf 6.58 0 Td[(3(n)]TJ /F4 11.955 Tf 11.96 0 Td[(2)]TJ /F3 11.955 Tf 11.96 0 Td[(i+1).Simplifyingthis,weconcludethatthenumberofmembersofPCa,v0forwhichz(D1))]TJ /F4 11.955 Tf 11.95 0 Td[(1=iisn)]TJ /F4 11.955 Tf 11.95 0 Td[(2i(i+1)i)]TJ /F5 7.97 Tf 6.59 0 Td[(1(n)]TJ /F3 11.955 Tf 11.95 0 Td[(i)]TJ /F4 11.955 Tf 11.95 0 Td[(1)n)]TJ /F15 7.97 Tf 6.59 0 Td[(i)]TJ /F5 7.97 Tf 6.59 0 Td[(3.HencewegetthatjPCa,v0j=n)]TJ /F4 11.955 Tf 11.95 0 Td[(20(n)]TJ /F4 11.955 Tf 11.95 0 Td[(1)n)]TJ /F5 7.97 Tf 6.59 0 Td[(3+n)]TJ /F4 11.955 Tf 11.95 0 Td[(21(n)]TJ /F4 11.955 Tf 11.95 0 Td[(2)n)]TJ /F5 7.97 Tf 6.58 0 Td[(4+n)]TJ /F13 5.978 Tf 5.75 0 Td[(3 2Xi=2n)]TJ /F4 11.955 Tf 11.96 0 Td[(2i(i+1)i)]TJ /F5 7.97 Tf 6.59 0 Td[(1(n)]TJ /F3 11.955 Tf 11.95 0 Td[(i)]TJ /F4 11.955 Tf 11.96 0 Td[(1)n)]TJ /F15 7.97 Tf 6.59 0 Td[(i)]TJ /F5 7.97 Tf 6.59 0 Td[(3.However,n)]TJ /F4 11.955 Tf 11.96 0 Td[(20(n)]TJ /F4 11.955 Tf 11.95 0 Td[(1)n)]TJ /F5 7.97 Tf 6.59 0 Td[(3=n)]TJ /F4 11.955 Tf 11.96 0 Td[(20(0+1)0)]TJ /F5 7.97 Tf 6.59 0 Td[(1(n)]TJ /F4 11.955 Tf 11.96 0 Td[(0)]TJ /F4 11.955 Tf 11.95 0 Td[(1)n)]TJ /F5 7.97 Tf 6.58 0 Td[(0)]TJ /F5 7.97 Tf 6.58 0 Td[(3,andn)]TJ /F4 11.955 Tf 11.96 0 Td[(21(n)]TJ /F4 11.955 Tf 11.95 0 Td[(2)n)]TJ /F5 7.97 Tf 6.59 0 Td[(4=n)]TJ /F4 11.955 Tf 11.96 0 Td[(21(1+1)1)]TJ /F5 7.97 Tf 6.59 0 Td[(1(n)]TJ /F4 11.955 Tf 11.96 0 Td[(1)]TJ /F4 11.955 Tf 11.95 0 Td[(1)n)]TJ /F5 7.97 Tf 6.58 0 Td[(1)]TJ /F5 7.97 Tf 6.58 0 Td[(3,sowemayincludethesetermsinthesum.Thatis, 52

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jPCa,v0j=n)]TJ /F13 5.978 Tf 5.76 0 Td[(3 2Xi=0n)]TJ /F4 11.955 Tf 11.96 0 Td[(2i(i+1)i)]TJ /F5 7.97 Tf 6.59 0 Td[(1(n)]TJ /F3 11.955 Tf 11.95 0 Td[(i)]TJ /F4 11.955 Tf 11.96 0 Td[(1)n)]TJ /F15 7.97 Tf 6.59 0 Td[(i)]TJ /F5 7.97 Tf 6.58 0 Td[(3.Byperformingthesubstitutionm=n)]TJ /F4 11.955 Tf 11.96 0 Td[(2,weobtainjPCa,v0j=m)]TJ /F13 5.978 Tf 5.76 0 Td[(1 2Xi=0mi(i+1)i)]TJ /F5 7.97 Tf 6.58 0 Td[(1(m)]TJ /F3 11.955 Tf 11.96 0 Td[(i+1)m)]TJ /F15 7.97 Tf 6.58 0 Td[(i)]TJ /F5 7.97 Tf 6.59 0 Td[(1,wherethelengthofv0isnowm+1.Abriefexaminationofthissumshouldthenconvincethereaderthatduetothesymmetricnatureoftheabovesum,wehavejPCa,v0j=m)]TJ /F5 7.97 Tf 6.59 0 Td[(1Xi=01 2mi(i+1)i)]TJ /F5 7.97 Tf 6.59 0 Td[(1(m)]TJ /F3 11.955 Tf 11.95 0 Td[(i+1)m)]TJ /F15 7.97 Tf 6.59 0 Td[(i)]TJ /F5 7.97 Tf 6.59 0 Td[(1.Fromtherewemayinvokearesultin[ 15 ]whichthengivesthatjPCa,v0j=(m+2)m)]TJ /F5 7.97 Tf 6.58 0 Td[(1=nn)]TJ /F5 7.97 Tf 6.59 0 Td[(3.Thuswehavethedesiredresultwhennisodd.Assumenowthatniseven.Ourcalculationshereareidenticaltothecasewherenisoddexceptincalculatingthelasttermofthesum.Sinceniseven,z(D1))]TJ /F4 11.955 Tf 13.08 0 Td[(1cantakeanyvalueinf0,1...,n)]TJ /F5 7.97 Tf 6.59 0 Td[(2 2g.IncountingthenumberofmembersofPCa,v0withagivenvalueofz(D1))]TJ /F4 11.955 Tf 12.67 0 Td[(1wegetthesameresultsaswhennisoddif0i=z(D1))]TJ /F4 11.955 Tf 12.15 0 Td[(1
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0@n)]TJ /F13 5.978 Tf 5.75 0 Td[(2 2)]TJ /F5 7.97 Tf 6.59 0 Td[(1Xi=0n)]TJ /F4 11.955 Tf 11.96 0 Td[(2i(i+1)i)]TJ /F5 7.97 Tf 6.59 0 Td[(1(n)]TJ /F3 11.955 Tf 11.95 0 Td[(i)]TJ /F4 11.955 Tf 11.96 0 Td[(1)n)]TJ /F15 7.97 Tf 6.59 0 Td[(i)]TJ /F5 7.97 Tf 6.58 0 Td[(31A+ 1 2)]TJ /F15 7.97 Tf 5.48 -4.32 Td[(n)]TJ /F5 7.97 Tf 6.58 0 Td[(2n)]TJ /F13 5.978 Tf 5.75 0 Td[(2 2n)]TJ /F4 11.955 Tf 11.96 0 Td[(2 2+1n)]TJ /F13 5.978 Tf 5.75 0 Td[(2 2)]TJ /F5 7.97 Tf 6.59 0 Td[(1n)]TJ /F3 11.955 Tf 13.15 8.08 Td[(n)]TJ /F4 11.955 Tf 11.95 0 Td[(2 2)]TJ /F4 11.955 Tf 11.96 0 Td[(1n)]TJ /F14 5.978 Tf 7.78 3.36 Td[(n)]TJ /F13 5.978 Tf 5.75 0 Td[(2 2)]TJ /F5 7.97 Tf 6.59 0 Td[(3!Againusingthesubstitutionm=n)]TJ /F4 11.955 Tf 11.88 0 Td[(2,andtakingadvantageofthesymmetryinthesum,wendthatjPCa,v0j=mXi=01 2mi(i+1)i)]TJ /F5 7.97 Tf 6.58 0 Td[(1(m)]TJ /F4 11.955 Tf 11.96 0 Td[(1+1)m)]TJ /F15 7.97 Tf 6.59 0 Td[(i)]TJ /F5 7.97 Tf 6.58 0 Td[(1.Againby[ 15 ]weconcludethatjPCa,v0j=(m+2)m)]TJ /F5 7.97 Tf 6.59 0 Td[(1=nn)]TJ /F5 7.97 Tf 6.59 0 Td[(3.Thusundertheassumptionthatv02Un)]TJ /F5 7.97 Tf 6.58 0 Td[(1,weconcludethatPCa,v0haspreciselynn)]TJ /F5 7.97 Tf 6.58 0 Td[(3members.WenowhavethatjPCa,v0j=nn)]TJ /F5 7.97 Tf 6.59 0 Td[(3=jHp,v0jjHa,v0j,and:Ha,v0!PCa,v0isinjective,soweconcludethat:Ha,v0!PCa,v0isalsosurjective.Sowehavethedesiredresultforeachv2Un)]TJ /F5 7.97 Tf 6.59 0 Td[(1.Anexampleisinorderhere.Unfortunatelyitisverydifculteithertodeterminethecriticallevelcurvecongurationofafunctionelement,ortondapolynomialwithagivencriticallevecurveconguration.Thereforeourexampleisquitesimple. Example: Considerthefunctionf(z)=1 .6z2+9 25ez.ThentheshadedregionGinFigure 5-1 isoneofthecomponentsofthesetfw:jf(w)j<1g,andtheboundaryofGisoneofthelevelcurvesoffonwhichjfj1.ThecriticalpointoffinGisatz=)]TJ /F4 11.955 Tf 9.29 0 Td[(.2.Consideralsothepolynomialp(z)=1 .6)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(z2+f()]TJ /F4 11.955 Tf 9.29 0 Td[(.2). 54

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TheshadedregionDinFigure 5-2 isthesetfw:jp(w)j<1g,andtheboundaryofDisoneofthelevelcurveofponwhichjpj1.Thecriticalpointofpisatz=0.ItiseasytoseethatthecriticalvaluewhicharisesfromthecriticalpointoffinGisequaltothecriticalvalueofp.SincethereisonlyonememberofPCwhichhasagivencriticalvalue,itfollowsthat(f,G)=(p,D).ThereforebyTheorem 4.1 ,thereissomeconformalfunction:G!DsuchthatfponG. 55

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Figure5-1. Tractoff Figure5-2. Tractofp 56

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CHAPTER6ISSURJECTIVE:THEGENERALCASEWenowwishtoshowthat:Ha,v0!PCa,v0issurjectiveforchoicesofv0inVnU.Thiswillalsobeaninductionargument,andweneedseveraldenitions. Denition27. Forv2Vn)]TJ /F5 7.97 Tf 6.59 0 Td[(1sayvistypicalifv2Un)]TJ /F5 7.97 Tf 6.58 0 Td[(1,inwhichcasewesayvhasatypicallitydegree0(so00,andanyv=(v(1),...,v(n)]TJ /F5 7.97 Tf 6.59 0 Td[(1))2Vn)]TJ /F5 7.97 Tf 6.59 0 Td[(1withatypicallitydegree0,andanyhiPC2PCa,v,thereissomeu2)]TJ /F5 7.97 Tf 6.58 0 Td[(1(v)suchthat(pu,Gpu)=hiPC. Denition28. Foru=(u(1),...,u(n)]TJ /F5 7.97 Tf 6.58 0 Td[(1))2Cn)]TJ /F5 7.97 Tf 6.58 0 Td[(1,deneapolynomialpubypu(w):=Zw0n)]TJ /F5 7.97 Tf 6.58 0 Td[(1Yi=1(z)]TJ /F3 11.955 Tf 11.96 0 Td[(u(i))dz.Thendene:Cn)]TJ /F5 7.97 Tf 6.58 0 Td[(1!Cn)]TJ /F5 7.97 Tf 6.59 0 Td[(1by(u)=(pu(u(1)),...,pu(u(n)]TJ /F5 7.97 Tf 6.58 0 Td[(1))).Ofcourseforanyu2Cn)]TJ /F5 7.97 Tf 6.58 0 Td[(1,thecriticalpointsofpuareexactlyu(1),...,u(n)]TJ /F5 7.97 Tf 6.58 0 Td[(1),somaybethoughtofastakingaprescribedlistofcriticalpointstoalistofcriticalvaluesviaanormalizedpolynomial(puisthepolynomialwithcriticalpointsu(1),...,u(n)]TJ /F5 7.97 Tf 6.58 0 Td[(1)normalizedsothatpu(0)=0andpu0ismonic).Thisfunctionwasstudiedin[ 14 ]inwhichitwasshownthatforanyn>0,andanyv=(v(1),...,v(n)]TJ /F5 7.97 Tf 6.58 0 Td[(1))2V,andany(p,Gp)2Hpwhosecriticalvaluesarev(1),...,v(n)]TJ /F5 7.97 Tf 6.59 0 Td[(1),thereissomeu2)]TJ /F5 7.97 Tf 6.58 0 Td[(1(v)suchthat(p,Gp)(pu,Gpu). Denition29. Foranyi0,letJ(i)denotethestatementForanyhiPC2PCawhosevectorofcriticalvaluesv2Vn)]TJ /F5 7.97 Tf 6.59 0 Td[(1hasatypicallitydegreelessthanorequaltoi,thereisau2)]TJ /F5 7.97 Tf 6.58 0 Td[(1(v)suchthat(pu,Gpu)2Hpand(pu,Gpu)=hiPC.WewishtoshowJ(i)holdsforeachi0.WehavealreadyshowninChapter 5 thatJ(0)holds.FixsomeM1andassumeinductivelythatJ(i)holdsforalli2f0,...,M)]TJ /F4 11.955 Tf 12.91 0 Td[(1g.WenowwishtoshowthatJ(M)holds.FixsomeN)]TJ /F4 11.955 Tf 12.91 0 Td[(1Mandsomev1=(v1(1),...,v1(N)]TJ /F5 7.97 Tf 6.59 0 Td[(1))2VN)]TJ /F5 7.97 Tf 6.59 0 Td[(1withatypicallitydegreeM.Assumethat 57

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v16=(0,...,0),sinceifthiswerethecasethenitiseasytoshowthedesiredresultholds.NowxsomememberhiPCofPCa,v1.OurplanistochooseanothermemberhbiPCofPCawhichisinsomesensetobedeterminedveryclosetohiPC,butwhoselistofcriticalvaluesbv1hasatypicallitydegreestrictlylessthanM.Bytheinductionassumption,thereissomebu12)]TJ /F5 7.97 Tf 6.59 0 Td[(1(bv1)suchthat(pbu1,Gpcu1)=hbiPC.IfwechoosehbiPCsothatbv1issufcientlyclosetov1,thiswillensurethatthemembersof)]TJ /F5 7.97 Tf 6.59 0 Td[(1(bv1)areclosetothemembersof)]TJ /F5 7.97 Tf 6.58 0 Td[(1(v1).Ifweletu1denoteamemberof)]TJ /F5 7.97 Tf 6.59 0 Td[(1(v1)whichbu1iscloseto,thenwewilshowthat(pu1,Gpu1)=hiPC.Firstacoupleofdenitions. Denition30. Fornon-zerox(1),x(2)2C,denedarg(x(1),x(2)):=8>><>>:jarg(x(1)))]TJ /F4 11.955 Tf 11.96 0 Td[(arg(x(2))j,ifx(1)6=06=x(2)andarg(x(1))6=arg(x(2))2,ifx(1)=0orx(2)=0orarg(x(1))=arg(x(2)),wherethechoiceofarg(x(1))andarg(x(2))inthedenitionaboveismadesoastominimizedarg(x(1),x(2)).Forx=(x(1),...,x(m))2Cm,denedarg(x):=min(darg(x(i),x(j)):1i,jm). Denition31. Fora,b2Rwitha
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thelimitasjIj!0ofjargj(f,,I)exists(althoughpossiblyinnite).Weletjargj(f,)denotethislimit,andwecalljargj(f,)thetotalvariationofarg(f)along. Denition32. Forx=(x(1),...,x(m))2Cm,deneminmod(x):=8>><>>:0,ifx(1)==x(m)=0min(jx(i)j:1im,x(i)6=0),otherwise. Denition33. Foranym2andr=(r(1),...,r(m))2Cm,denemindiff(r(1),...,r(m)):=8>><>>:0,ifr(1)==r(m)min(jr(i))]TJ /F3 11.955 Tf 11.95 0 Td[(r(j)j:r(i)6=r(j),1i0smallenoughthatthefollowinghold. 1. Letu=(u(1),...,u(N)]TJ /F5 7.97 Tf 6.59 0 Td[(1))besomepointin)]TJ /F5 7.97 Tf 6.59 0 Td[(1(v1).Fixsomei2f1,...,N)]TJ /F4 11.955 Tf 12.53 0 Td[(1gsuchthatv1(i)6=0,andletbu2B1(u)begiven.LetLbealinesegmentcontainedinB41(u(i)).Thenjargj(pbu,L)
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thissectionofgradientlinewhichconnectsz(i)andy(i).Since(i)isaportionofagradientlineofpu,arg(pu(y(i)))=arg(pu(z(i))),soy(i)isnotacriticalpointofpu.Sincethereareonlynitelymanysuchu,,andD,wemayconstructsuchacollectionofpathsforeachsuchchoiceofu,,D,andchoose1sothatforeachsuchu,,andD,ifi2f1,...,mg(mdependsonthechoiceofu,,andD)andt2[0,1],thereisnoj2f1,...,mgnfigands2[0,1]suchthat(j)(s)iswithin21of(i)(t),andnocriticalpointofpuiswithin21of(i)(t),andthereisnoedgeofanycriticallevelcurveofpuotherthanE(i)within21of(i)(t). 4. Foreachu2)]TJ /F5 7.97 Tf 6.58 0 Td[(1(v1),nocriticallevelcurveofpuiswithin21of@Gpu,andnocriticallevelcurveofpuiswithin21ofanyzeroofpu,andnocriticallevelcurveofpuiswithin31ofanyothercriticallevelcurveofpu. 5. Foreachu2)]TJ /F5 7.97 Tf 6.59 0 Td[(1(v1),andeachk2f1,...,N)]TJ /F4 11.955 Tf 12.25 0 Td[(1g,thereisnopointB2(1)(u(k))nfu(k)gatwhichputakesthevaluev1(k). 6. Foreachu2)]TJ /F5 7.97 Tf 6.59 0 Td[(1(v1),ifjbu)]TJ /F3 11.955 Tf 12.41 0 Td[(uj<1,thenforeachk2f1,...,N)]TJ /F4 11.955 Tf 12.41 0 Td[(1gsuchthatv1(k)6=0,ifjz)]TJ /F3 11.955 Tf 11.95 0 Td[(u(k)j<21,thenjpbu(z)j>minmod(v1) 2. 7. 121.Wenowchoose2>0smallenoughsothateachofthefollowingholds. 1. 2<1. 2. Foreachu2)]TJ /F5 7.97 Tf 6.59 0 Td[(1(v1),foreachk2f1,...,N)]TJ /F4 11.955 Tf 12.06 0 Td[(1g,foreachz2B32(u(k)),wehavejpu(u(k)))]TJ /F3 11.955 Tf 11.96 0 Td[(pu(z)j<1. 3. ByLemma 10 ,wemaychoose2>0and1>0sothatforanyu2)]TJ /F5 7.97 Tf 6.58 0 Td[(1(v1),letbubeanypointinB1(u)andletbx1,bx22Gpbubegivensuchthatarg(pu(x1))=arg(pu(x2))=0,andsuchthatthereisapathb:[0,1]!Gpusuchthatb(0)=bx1andb(1)=bx2andarg(pbu(b(r)))=0forallr2[0,1].Thenifx1,x22Gpbuaresuchthatarg(pu(x1))=arg(pu(x1))=0andjbx1)]TJ /F3 11.955 Tf 12.18 0 Td[(x1j<2andjbx2)]TJ /F3 11.955 Tf 12.19 0 Td[(x2j<2,thenthereisapath:[0,1]!Gpusuchthat(0)=x1,(1)=x2,andforallr2[0,1],arg(pu((r)))=0andjb(r))]TJ /F6 11.955 Tf 12.61 0 Td[((r)j<1.Moreover,ifjpbujisstrictlyincreasingorstrictlydecreasingalongb,thenmaybechosensothatjpujisincreasingordecreasingalongrespectively.InItem 3 abovewechosea1>0.Wenowrequirethat1>0bechosensothatthefollowingholds. 60

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1. 1<2. 2. Wewillusethisseconditemtorefertotherestrictionon1describedinItem 3 forthechoiceof2above. 3. Letu=(u(1),...,u(N)]TJ /F5 7.97 Tf 6.59 0 Td[(1))2)]TJ /F5 7.97 Tf 6.58 0 Td[(1(v1)bechosen.Forbu2B1(u)denebv=(dv(1),...,\v(n)]TJ /F5 7.97 Tf 6.58 0 Td[(1)):=(bu).Supposethatarg(dv(k))=arg(v(k))foreachk2f1,...,N)]TJ /F4 11.955 Tf 12.92 0 Td[(1g.Forsomek2f1,...,N)]TJ /F4 11.955 Tf 12.92 0 Td[(1gwithjv(k)j6=0,letbdenotethelevelcurveofpbuwhichcontainsdu(k).Thenthefollowingholds.LetbEdenotesomeedgeofbwhichisincidenttodu(k),andletbdenoteaparameterizationofbEbeginningatdu(k)parameterizedwithrespecttoarg(pbu).Thatis,ifisthetotalchangeinargumentofarg(pbu)alongbE,and2[0,2)istheargumentofdv(k),thenb:[,+]!bandsatisesb()=du(k)andarg(pbu(b(t)))=tforallt2[,+].Thenifweletdenotethecriticallevelcurveofpucontainingu(k),thereisapath:[,+]!suchthat()=u(k),andforeachr2[,+],arg(pu((r)))=randj(r))]TJ /F18 11.955 Tf 12 0 Td[(b(r)j<2.ThismaybedonebyLemma 9 4. Foreachu2)]TJ /F5 7.97 Tf 6.59 0 Td[(1(v1),foreachi2f1,...,N)]TJ /F4 11.955 Tf 11.95 0 Td[(1g,B1(u(i))Gpu. 5. Foreachu2)]TJ /F5 7.97 Tf 6.59 0 Td[(1(v1),ifjbu)]TJ /F3 11.955 Tf 12.91 0 Td[(uj<1,andz2Gpbu,andjz)]TJ /F3 11.955 Tf 12.91 0 Td[(z0j<1,thenjpbu(z0))]TJ /F3 11.955 Tf 11.96 0 Td[(pu(z)j<2. 6. 10smallsothatthefollowingholds. 1. 1<1. 2. Ifbv12VN)]TJ /F5 7.97 Tf 6.59 0 Td[(1satisesjv1)]TJ /F18 11.955 Tf 11.5 .5 Td[(bv1j<1,andbu2)]TJ /F5 7.97 Tf 6.58 0 Td[(1(bv1),thenthereissomeu2)]TJ /F5 7.97 Tf 6.58 0 Td[(1(v1)suchthatju)]TJ /F18 11.955 Tf 12.02 .49 Td[(buj<1 4.ThismaybedonebyLemma 5 3. 1
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Denition34. Foreachnon-zerointegerk,lethwkiPdenotethesinglepointmemberofPsuchthatZ(hwkiP)=k. Denition35. ForhiPC2PC,and>0wedeneEhiPC,tobethecollectionofmembersh iP2PusedtoconstructhiPCsuchthatH(h iP)=.WewillconstructhbiPC2PCdifferentlydependingonwhichofthefollowingthreecasesintowhichhiPC2PCfalls. jv1(M)j=0. jv1(M)j>0andforeachhiP2EhiPC,jv1(M)j,hiPonlycontainsasinglevertex(countingmultiplicity). jv1(M)j>0andthereissomememberofEhiPC,jv1(M)jwhichcontainsmorethanonevertex(countedwithmultiplicity). Case6.0.6. jv1(M)j=0.Since0isacriticalvalueofhiPC,thereissomelevel0memberhwkiPC2PCusedintheconstructionofhiPCsuchthatk2.Thatis,intheconstructionofhiPC,hwkiPCwasassociatedtoafaceofsomememberofP.Leth iPdenotethismemberofP,andletDdenotethefaceof towhichhwkiPCwasassociated.ThengDmappedeachdistinguishedpointin@Dtowk.WewilldenehbiPC,anothermemberofPC,toreplacehwkiPCasweconstructhbiPC,andineveryotherrespectweconstructhbiPCinthesamemannerashiPC.Letbdenotethegureeightplanargraph.Letxdenotethevertexofb.DeneH(hbiP):=1 2,anda(x):=0.LetD(1)denoteoneoftheboundedfacesofb,andD(2)theother.Distinguishxanddistinguishk)]TJ /F4 11.955 Tf 12.2 0 Td[(2distinctpointsotherthanxintheboundryofD(1).WiththisauxiliarydatawehaveformedamemberofP,namelyhbiP.ToD(1)weassociatehwk)]TJ /F5 7.97 Tf 6.58 0 Td[(1iPC,anddenegD(1)bymappingeachdistinguishedpointin@D(1)towk)]TJ /F5 7.97 Tf 6.58 0 Td[(1,andassociatehw1iPCtoD(2),anddenegD(2)tomapthesingledistinguishedpointin@D(2)(namelyx)tow1.TheresultingobjectisamemberofPC,namelyhbiPC. 62

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WewishtoconstructhbiPCinexactlythesamemannerashiPC,exceptbyreplacinghwkiPCwithhbiPC.Wemaydothisbecausebyconstruction,Z(hbiP)=k=Z(hwkiP).TheonlythingremainingtodointheconstructionofhbiPCisspecifygD.Letw(1)beanyxeddistinguishedpointin@D.ThendenegD(w(1)):=x,andifwistheithdistinguishedpointin@D(forsomei2f1,...,k)]TJ /F4 11.955 Tf 11.97 0 Td[(1g)inthepositivedirectionafterw(1),denegD(w)tobetheithdistinguishedpointin@D(1)inthepositivedirectionafterx(wherethe(k)]TJ /F4 11.955 Tf 12.05 0 Td[(1)stdistinguishedpointin@D(1)afterxisinterpretedasbeingxitself).ThenproceedingwiththeconstructionineveryotherwaythesameaswithhiPC,weobtainamemberofPC,namelyhbiPC.NotethatthecritcalvaluesofhbiPCwillbeexactlybv1:=(0,...,0,1 2,v1(M+1),...,v1(N)]TJ /F5 7.97 Tf 6.58 0 Td[(1))(withM)]TJ /F4 11.955 Tf 12.16 0 Td[(1copiesof0),whilev1=(0,...,0,v1(M+1),...,v1(N)]TJ /F5 7.97 Tf 6.59 0 Td[(1)),sojv1)]TJ /F18 11.955 Tf 14.04 .49 Td[(bv1j=1 2<1.Notealsothatsince1 20andforeachhiP2EhiPC,jv1(M)j,hiPonlycontainsasinglevertex(countingmultiplicity).LethiPbesomexedmemberofEhiPC,jv1(M)j.WeconstructhbiPCidenticallytotheconstructionofhiPC,exceptwereplacehiPCintheconstructionwithhbiPC,wherehbiPCisidenticaltohiPCexceptthatH(hbiP):=(1+1 2)H(hiP)=(1+1 2)jv1(M)j.Notethatwiththisconstruction,thecriticalvaluesofhbiPCareexactlybv1:=(v1(1),...,v1(M)]TJ /F5 7.97 Tf 6.59 0 Td[(1),(1+1 2)v1(M),v1(M+1),...,v1(N)]TJ /F5 7.97 Tf 6.59 0 Td[(1)),sojv1)]TJ /F18 11.955 Tf 13.86 .5 Td[(bv1j=j1 2v1(M)j1 2<1.Foreachk2f1,...,N)]TJ /F4 11.955 Tf 11.95 0 Td[(1g,letdv1(k)denotethekthentryofbv1.Thenj\v1(M)]TJ /F5 7.97 Tf 6.59 0 Td[(1)j=jv1(M)]TJ /F5 7.97 Tf 6.59 0 Td[(1)j=jv1Mj
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j\v1(M+1)j)-222(j[v1(M)j1)]TJ /F6 11.955 Tf 13.15 8.08 Td[(1 2jv1(M)j>1 2>0.Thusweconcludethatbv1hasatypicallitydegreelessthanM. Case6.0.8. jv1(M)j>0andsomememberofEhiPC,jv1(M)jcontainsmorethanonevertex(countingmultiplicity).LethiPCdenoteoneofthemembersofPCusedinconstructinghiPCsuchthathiP2EhiPC,jv1(M)j,andsuchthatcontainsmorethanonevertex.(PossiblyhiPC=hiPC.)WenowconstructhbiPCwhichwilltaketheplaceofhiPCasweconstructhbiPC.Firstadenition. Denition36. LethiPC2PCabegivenwiththeassumptionthathasmorethantwoedges.ByLemma 2 ,wemayndsomeboundedfaceFofsuchthattheboundaryofFconsistsofasingleedgeEof. WedenenEtobethememberofPwhicharisesfromthegraphwhentheedgeEisremoved. WedenehnEiPtobethememberofPwhicharisesfromnE,andinheritsallofitsauxiliarydatafromhiP.NotethatifxisthevertexofwhichEhasasitsendpoints,ifthemultiplicityofxasavertexofequals1,thenxisnolongeravertexofnE,andthusa(x)nolongerhasanymeaningforhnEiP. WedenehnEiPCtobethememberofPwhicharisesfromnE,andinheritsallofitsauxiliarydatafromhiPC.Forexample,ifDisaboundedfaceofotherthanF,andhiPCisthememberofPCassociatedtoDinhiPC,thenweassociatehiPCtoDintheconstructionofhnEiPC,andwecarryovergDtohnEiPCaswell.ByLemma 2 ,thereissomefaceofwhichhasonlyoneedgeofinitsboundary.LetF(1),...,F(h)betheboundedfacesof,orderedsothatF(h)hasonlyoneedgeofinitsboundry,andletEdenotetheedgethatformstheboundryofF(h).LetzdenotethevertexofwhichisincidenttoF(h).Letmdenotethemultiplicityofzasavertexof.Asnotedabove,ifm=1,thenzisnotavertexofnE(oronemightsaythatzisa 64

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vertexofnEwithmultiplicity0),whileifm>1thenzisavertexofnEwithmultiplicitym)]TJ /F4 11.955 Tf 11.96 0 Td[(1.NotethatZ(hnEiP)=h)]TJ /F5 7.97 Tf 6.59 0 Td[(1Xk=1z(F(k)).Letbdenotethegureeightgraph,andletD(1),D(2)denoteitstwofaces.Letxdenotethevertexinb.FrombwewillformamemberofP,andeventuallyamemberofPCwhichwillreplacehiPCintheconstructionofhiPC.DeneH(hbiP):=(1+1 2)H(hiP).Denez(D(1)):=Z(hnEiP),andz(D(2)):=z(F(h))(whereF(h)isviewedasafaceof).Distinguishz(D(i))pointsin@D(i)fori=1,2,distinguishingxifandonlyifzisdistinguishedasavertexinhiP.Denea(x):=a(z)wherea(z)comesfromhiP.Withthisdata,wehaveamemberofP,namelyhbiP.LethiPCbethememberofPCwhichwasassociatedtoF(h)intheconstructionofhiPC.ThenweassociatehnEiPCtoD(1),andhiPCtoD(2).WenowwishtodenegD(1)andgD(2).InordertodenegD(1),wedeneanenumerationofthedistinguishedpointsinnE.LetE0denotetheedgeinwhichisdirectlyafterEasistraversedwithapositiveorientation.Deney(1):=zifzisadistinguishedpointinnE.Otherwisedeney(1)tobetherstdistinguishedpointafterzinnEasnEistraversedwithapositiveorientationbeginningwithE0.ContinuetraversingnEwithapositiveorientation,andlety(2),...,y(z(D(1)))bethedistinguishedpointsaftery(1)ofnEastheyappearasnEistraversedonefulltimebeginningwithE(1).NotethatadistinguishedpointwillappearinthislistexactlyntimesifitisadistinguishedpointofnEwithmultiplicityn.Nowletx(1),...,x(z(D(1)))beanenumerationofthedistinguishedpointsin@D(1)astheyappearinincreasingorderbeginningwithxifxisadistinguishedpoint,andbeginningwiththerstdistinguishedpointafterxotherwise.FinallywedenegD(1)(x(i)):=y(i)foreachi.WewillnowdenegD(2).Nowlety(1),...,y(z(D(2)))bethedistinguishedpointsin@F(h)listedinincreasingorder,withy(1)=zifzisdistinguishedinhiP,andy(1)therstdistinguishedpointafterzin@F(h)otherwise.Letx(1),...,x(z(D(2)))bethedistinguishedpointsin@D(2)listedinincreasingorder,withx(1)=xifxisdistinguished,andx(1) 65

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therstdistinguishedpointinthepositivedirectionfromxin@D(2)otherwise.Thenfor1iz(D(2)),wedenegD(2)(x(i)):=gF(h)(y(i)).WiththisdatawenowhaveamemberofPC,namelyhbiPC.WenowwishtoconstructhbiPCineveryrespectthesameashiPC,exceptthathiPCwillbereplacedinthisconstructionwithhbiPC.IfhiPC=hiPC,thenwearedone,andwedenehbiPC:=hbiPC.IfhiPC6=hiPC,thenhiPCwasassociatedtosomefaceDofh iPamemberofPduringtheconstructionofhiPC.ThenhbiPCmayinheritallofitsdatafromhiPC(otherthanhiPC,whichwehaveexchangedforhbiPC)exceptthegradientfunctiongD.LetgDdenotethegradientmapforDinhiPC(whichmapsthedistinguishedpointsin@Dtothedistinguishedpointsin),andletcgDdenotethegradientmapforDinhbiPC(whichwillmapthedistinguishedpointsin@Dtothedistinguishedpointsinb).ToconstructcgD,wehavetwopossiblecases,rstthatzisadistinguishedpointin,and,infact,theonlydistinctdistinguishedpointin,andsecondthattherearedistinguishedpointsinwhicharedistinctfromz. Subcase6.0.8.1. zisadistinguishedpointin,andzistheonlydistinctdistinguishedpointin.Letx(1),...,x(z(D))beanenumerationofthedistinguishedpointsin@Dlistedinincreasingorder.Lety(1),...,y(z(D))beanenumerationofthedistinguishedpointsinblistedintheorderinwhichtheyappearasbistraversed,beginningandendingwithx.ThenwedenecgD(x(i)):=y(i)foreachi. Subcase6.0.8.2. Therearedistinguishedpointsinwhicharedistinctfromz.WedeneanenumerationofthedistinguishedpointsofwhichwillbeusedtodenecgD.Letdy(1),...,\y(z(D(1)))beanenumerationofthedistinguishedpointsofbwhicharein@D(1),beginningatxifxisadistinguishedpointofb,andbeginningattherstdistinguishedpointpastxotherwise.Foreachi2f1,...,z(D(1))g,gD(1)(cy(i))isadistinguishedpointinnE.Deney(i)tobethedistinguishedpointinwhichcorrespondstothedistinguishedpointgD(1)(cy(i))innE.Fori2f1,...,z(D(2))g, 66

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lety(z(D(1))+i)denotetheithdistinguishedpointin@F(h)inincreasingorder,beginningwithzifzisdistinguished,andbeginningwiththerstdistinguishedpointpastzotherwise.Let\y(z(D(1))+i)denotetheithdistiguishedpointin@D(2),beginningwithxifxisadistinguishedpointofb,andwiththerstdistinguishedpointpastxinincreasingorderaround@D(2)otherwise.Thenfy(1),...,y(z(D(1))+z(D(2)))gisanenumerationofthedistinguishedpointsininincreasingorder,andfdy(1),...,\y(z(D(1))+z(D(2)))gisanenumerationofthedistinguishedpointsinbinorderastheyappeararoundb.Letz(1),...,z(z(D(1))+z(D(2)))beanyenumerationofthedistinguishedpointsin@DsuchthatgD(z(i))=y(i)foreachi2f1,...,z(D(1))+z(D(2))g.Nowfori2f1,...,z(D(1))+z(D(2))g,wedenecgD(z(i)):=cy(i).Withthisdenition,wehaveallthedataneededforamemberofPC,namelyhbiPC.Notice,then,thatbytheconstructionofhbiPC,thecriticalvaluesofhbiPCareexactlybv1:=(v1(1),...,v1(M)]TJ /F5 7.97 Tf 6.59 0 Td[(1),(1+1 2)v1(M),v1(M+1),...,v1(N)]TJ /F5 7.97 Tf 6.59 0 Td[(1))2VN)]TJ /F5 7.97 Tf 6.59 0 Td[(1.ThereforebythesameargumentasinCase 6.0.7 ,bv1hasatypicallitydegreelessthanM,andjv1)]TJ /F18 11.955 Tf 13.84 .5 Td[(bv1j=1 2jv1(M)j<1 2<1.SonowinanycasewehaveamemberofPC,hbiPC,withcriticalvaluesbv1suchthatjv1)]TJ /F18 11.955 Tf 14.09 .5 Td[(bv1j<1andtheatypicallitydegreeofbv1isstrictlylessthanM.Notealsothatbyconstruction,foreachk2f1,...,N)]TJ /F4 11.955 Tf 12.18 0 Td[(1g,arg(dv1(k))=arg(v1(k))(whereweadopttheconventionthatarg(0)=0).Bytheinductionassumptionthereissomebu1=(du1(1),...,\u1(N)]TJ /F5 7.97 Tf 6.59 0 Td[(1))2)]TJ /F5 7.97 Tf 6.58 0 Td[(1(bv1)suchthat(pbu1,Gpcu1)=hbiPC.ByItem 2 inthechoiceof1,thereisau1=(u1(1),...,u1(N)]TJ /F5 7.97 Tf 6.59 0 Td[(1))2CN)]TJ /F5 7.97 Tf 6.58 0 Td[(1suchthat(u1)=v1andju1)]TJ /F18 11.955 Tf 14.2 .5 Td[(bu1j<1.DeneheiPC:=(pu1,Gpu1).OurgoalistoshowthathiPC=heiPC.Firsttwodenitions. 67

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Denition37. LethiPCbesomememberofPC,andletDdenotesomefaceof.ThenwelethDiPCdenotethememberofPCwhichwasassociatedtoDintheconstructionofhiPC. Denition38. LethiPbeagraphmemberofP,andletn2denotethenumberofedgesof.WesayanenumerationE(1),...,E(n)oftheseedgesisinorderwithrespectto(orjustinorderwhenisobvious)iftheorderinwhichtheedgesappearwhenistraversedonefulltimewithpositiveorientationbeginningwithE(1)isexactlyE(1),...,E(n).LetDbeaboundedfaceof,andletk1denotethenumberofedgesofwhicharein@D.WesaythatanenumerationE(1),...,E(k)oftheseedgesisinorderwithrespecttoDifE(1),...,E(k)istheorderinwhichtheseedgesappearas@DistraversedonefulltimewithpositiveorientationbeginningwithE(1).WewillshowthathiPC=heiPCrecursively,workingoutsidein,bydoingthefollowingsteps. 1. ShowthathiP=heiP,andestablishacorrespondencebetweentheverticesofhiPandtheverticesofheiPwhichrespectsthedatacontainedinamemberofP.(Thatis,ifk1isthenumberofverticesinande,ndenumerationsu(1),...,u(k)andgu(1),...,gu(k)oftheverticesofanderespectivelysuchthatthefollowingholds.Foreachi2f1,...,kg,a(u(i))=a(fu(i)).Foreachi,j2f1,...,kg,fu(i)u(j)gisanedgeinifandonlyifffu(i)fu(j)gisanedgeineand,moreover,iffu(i)u(j)gisanedgein,thenfu(i)u(j)gandffu(i)fu(j)gcontainthesamenumberofdistinguishedpoints.Finally,ifn2isthenumberofedgesin,andfu(i1)u(i2)g,...,fu(in)u(in+1)gisthelistofedgesofastheyappearinorderaround,thenfgu(i1)gu(i2)g,...,fgu(in)^u(in+1)gistheorderinwhichtheedgesofeappeararounde.NotethatthiswillimmediatelyprovideawelldenedcorrespondencebetweentheboundedfacesDofandtheboundedfaceseDofeandbetweenthedistinguishedpointsxofhiPandthedistinguishedpointsexofheiP.) 2. LetDbeoneoftheboundedfacesof,andlethDiPCandhfeDiPCdenotethemembersofPCassignedtoDandeDduringtheconstructionofhiPCandheiPCrespectively.ShowthathDiP=hfeDiP,andestablishacorrespondencebetweentheverticesofDandtheverticesoffeDasdescribedinStep 1 .ThenshowthatthecorrespondencebetweenhDiPandhfeDiPrespectsthegradientmapsgDandgeD.Thatis,ifxisoneofthedistinguishedpointsofin@D,andexis 68

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thecorrespondingdistinguishedpointofein@eD,thenshowthatgeD(ex)isthedistinguishedpointinfeDwhichcorrespondstothedistinguishedpointgD(x)inD. 3. IterateStep 2 foreachfaceDof,thenagainwitheachfaceofeachD,etc.SincehiPC,heiPCareconstructedwithnitelymanysteps,thisprocesswillterminateafternitelymanysteps.Whenthisprocessterminates,wewillhaveshownthathiPCandheiPChaveallthesamedata,andarethereforeequal.Noticethatthebasecase(Step 1 andStep 2 aswritten)isjustasimplercaseoftherecursivestep(Step 1 andStep 2 withanyhiPCusedintheconstructionofhiPCinsertedintheplaceofhiPC).Thereforewejustdotherecursivestep.NowforanyhiPCusedtoconstructhiPC,wewillseethatintheprocessofestablishingthecorrespondenceoftheedgesandverticesdescribedabovebetweenhiPCandthecorrespondingheiPCusedtoconstructheiPCwewilldothefollowing.LetDbeafaceof.LeteDbethecorrespondingfaceofe.WewillselectafacebDofsomegraphmemberhbiPCusedtoconstructhbiPC,suchthatbDcorrespondsnaturallytoDintheconstructionofhbiPC.WewillvieweandbasembeddedinCascriticallevelcurvesofpu1andpbu1respectively,andndpathsandbwhichparameterizeeDandbDrespectivelysuchthatj(r))]TJ /F18 11.955 Tf 12.89 0 Td[(b(r)j<1foreachr.Thisthenimpliesthatforanyi2f1,...,N)]TJ /F4 11.955 Tf 12.36 0 Td[(1g,ifu1(i)2eD,thendu1(i)2bD.Toseethisimplication,observethat,byItem 5 inthechoiceof1,ifu1(i)2eD,thenB31(u1(i))eD.Butju1(i))]TJ /F18 11.955 Tf 14.21 4.28 Td[(du1(i)j<1<1,soB21(du1(i))eD.Sincej(r))]TJ /F18 11.955 Tf 12.21 0 Td[(b(r)j<1forallr,thewindingnumberofbarounddu1(i)isthesameasthewindingnumberofaroundu1(i).Thusweconcludethatdu1(i)2bD.Nowletusreturntoourinductionargument.SupposethatStep 2 hasjustbeencompletedforsomehiPusedtoconstructhiPC,withcorrespondingheiPusedtoconstructheiPC.LetDbeoneoftheboundedfacesof,andleteDbethecorrespondingboundedfaceofe.WewillnowdescribetheselectionofthefacebDreferredtoabove.LethbiPCdenotethememberofPCwhichreplacedhiPCintheconstructionofhbiPC. 69

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Case6.0.9. hbiP=hiP.ThenletbDdenotethefaceDofviewedasafaceofb. Case6.0.10. hbiP6=hiP.TheneitherhbiPwasformedbymerelychangingthevalueofH()(thatis,hbiP=hiPexceptthatH(hbiP)=(1+1 2)H(hiP))orthegraphitselfwaschangedtoformhbiPC.IfhbiPwasformedbymerelychangingthevalueofH(),thenwemaychoosebDtobethefaceDofviewedasafaceofbasinthepreviouscase.SupposethatitselfwaschangedwhileforminghbiPC.Recallthatinthiscase,hbiPCwasformedbyselectingaboundedfaceFofsuchthat@FconsistsofonlyasingleedgeE(1)of.Wethensetbtobethegureeightgraph,andassignhFiPCtoonefaceofb,andhnE(1)iPCtotheotherfaceofb.Wedeneauxiliarydata(valuesofa(),valuesofH(),distinguishedpoints,gradientmaps)asdescribedearlierinthischapter,andtheresultingmemberofPCwecallhbiPC.(FromnowonwewillrefertohnE(1)iPCash\nE(1)iPCtoremindusthatweareviewingh\nE(1)iPCasamemberofPCusedtoconstructhbiPC.)LetuscallthismethodofforminghbiPCjustdescribedthescatteringmethod,asitscattersoneoftheverticesof.Withtheabovedescription,ifD=FwedenebDtobethefaceofbtowhichhFiPCwasassigned.IfD6=F,wedenebDtobethefaceof\nE(1)whichcorrespondstoDintheconstructionofh\nE(1)iPC.WenowlethDiPC,hcbDiPC,andhfeDiPCbethemembersofPCwhichareassignedtoD,bD,andeDintheconstructionofhiPC,hbiPC,andheiPCrespectively.Beforewediveintothenextargument,letusstepbackforamomentandreviewourstrategy.hiPCwasgiventousasamemberofPCwiththeprescribedcriticalvaluesv1,andourgoalistondapolynomialpsuchthat(p,Gp)=hiPC.WeconstructedhbiPCinsuchawayastobeinsomesenseverysimilartohiPC,havecriticalvaluesbv1veryclosetov1,andsothat,bytheinductionassumption,thereissomebu12CN)]TJ /F5 7.97 Tf 6.59 0 Td[(1suchthat(bu1)=bv1and(pbu1,Gpcu1)=hbiPC.WethenusedLemma 5 tondapointu12CN)]TJ /F5 7.97 Tf 6.58 0 Td[(1 70

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closetobu1andsuchthat(u1)=v1.WedeneheiPC:=(pu1,Gpu1).WeviewhbiPCandheiPCasbeingembeddedinCasthecriticallevelcurvesofpbu1andpu1respectively,andwewishtousethefactthatbu1andu1aresoclose,alongwiththefactthathiPCandhbiPCareclose(insomesense),toshowthathiPC=heiPC=(pu1,Gpu1),whichisourdesiredresult.RightnowwewishtoshowthathDiP=hfeDiP. Case6.0.11. hcbDiP6=hDiP,andhcbDiPCwasformedusingthescatteringmethod.LetL2denotethenumberofedgesinD.WewillagainletFdenotetheboundedfaceofDwhichwasremovedduringtheformationofhcbDiPC,andletbFdenotethefaceofcbDtowhichweassignedhFiPC.LetbGdenotetheotherfaceofcbD,namelytheonetowhichh\DnE(1)iPCwasassigned.RecallthatE(1)istheedgeofDwhichformstheboundaryofFandletE(2),...,E(L)betheenumerationoftheotheredgessothattheorderinwhichtheedgesofDappearinorderaroundDbeginningwithE(1)isexactlyE(1),...,E(L).LetK1denotethenumberofdistinctverticesofD,andletx(1),...,x(K)beanyenumerationoftheseverticessuchthatE(1)hasbothitsendpointsatx(1).Nowforeachi2f1,...,Lg,wewishtochooseanedge(oraportionofanedge)ofagraphusedtoconstructhcbDiPCwhichwillcorrespondtotheedgeE(i)inD. Subcase6.0.11.1. i=1.RecallthatE(1)formstheboundaryofthefaceFofD.InthiscasewedenedE(i)tobetheedgeofcbDwhichformstheboundaryofbF. Subcase6.0.11.2. i6=1,andx(1)isnotanendpointofE(i).InthiscasewedenedE(i)tobetheedgeE(i)viewedasanedgeof\DnE(1). Subcase6.0.11.3. i6=1,andx(1)isanendpointofE(i).Thedifcultyinthiscasearisesfromthefactthatifx(1)isincidenttoonlytwoboundedfacesofD,thenx(1)isnolongeravertexof\DnE(1). Subcase6.0.11.4. TherearemorethantwoboundedfacesofDwhichareincidenttox(1). 71

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Thenx(1)isstillavertexofof\DnE(1),sowemaydenedE(i)tobetheedgeE(i)viewedasanedgeinthegraph\DnE(1). Subcase6.0.11.5. ThereareonlytwoboundedfacesofDwhichareincidenttox(1).Firstadenition. Denition39. LethiPbeagraphmemberofP,andletEbeanedgeof.LetxdenotetheinitialpointofEandletydenotethenalpointofE.WewishtodeneaquantitywhichwewillcallthechangeinargumentalongE(withrespecttohiP).Dener1:=a(x)anddener2:=a(y).Ifa(y)=0,thenweinsteaddener2:=2.ThenwedenethechangeinargumentalongEwithrespecttohiPtober2)]TJ /F3 11.955 Tf 12.38 0 Td[(r1+2n,wherendenotesthenumberofdistinguishedpointsinEwhicharenotendpointsofE.NotethatifhiParisesasacriticallevelcurveofsomeanalyticfunctionf,thenthechangeinargumentalongEwithrespecttohiPisthesameasthechangeinarg(f)alongE.Letj2f1,...,LgbetheindexoftheotherendpointofE(i).Recallthatinthissub-casex(1)isoneendpointofE(i).RecallthatFisoneoftheboundedfacesofDwhichisincidenttox(1).LetGdenotetheother.If@GisformedbyasingleedgeofD,thenDisthegureeightgraph.HoweversincecbDwasformedusingthescatteringmethod,Disnotthegureeightgraph,so@GcontainsmorethanoneedgeofD.SinceE(i)doesnotformtheboundaryofF,weconcludethatE(i)iscontainedin@G.ThereforeE(i)doesnothavebothendsatx(1),andthereforej6=1.Thereforealsowemayviewx(j)asavertexinDnE(1).Let>0denotethechangeinargumentalongE(i)whereE(i)istraversedfromx(j)tox(1).LetbEdenotetheedgeof\DnE(1)whichcontainsthepointx(1)(whichisnolongeravertexof\DnE(1)).ThenletzdenotethepointinbEsuchthatthechangeinarg(pbu1)alongtheportionofbEbeginningatx(j)andendingatzisexactly.ThenwedenedE(i)tobethisportionofbE.NotethatineachcasebytheconstructionofhbiPC,thechangeinargumentalongdE(i)isthesameasthechangeinargumentalongE(i). 72

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Wenowwishforeachi2f1,...,Kgtochooseavertexcx(i)ofoneofthegraphsusedtoconstructhcbDiPCinsuchawaythatx(i)givesrisetocx(i)duringtheconstructionofhbiPC.Letdx(1)denotethevertexincbDand,foreachi2f2,...,Kg,letcx(i)denotethevertexx(i)viewedasavertexof\DnE(1).WenowhaveanameforthevertexincbDandforeachvertexin\DnE(1)viewedasagraphusedtoformhbiPC,unlessx(1)isstillavertexof\DnE(1).Inthiscase,letdx(0)denotethevertexx(1)in\DnE(1).NowviewhbiPCasembeddedinCasthecriticallevelcurvesofpbu1.Foreachi2f1,...,Kg,xsomechoiceofti2f1,...,N)]TJ /F4 11.955 Tf 12.7 0 Td[(1gsuchthat[u1(ti)isthecriticalpointofpbu1whichgivesrisetocx(i)intheconstructionof(pbu1,Gpcu1).Thenwedenefx(i):=u1(ti)(notethatthisimpliesthatt1=MbytheconstructionofhbiPC).Ifdx(0)isdened,thenlett02f1,...,N)]TJ /F4 11.955 Tf 12.15 0 Td[(1gbesomechoiceofindexsothat[u1(t0)isthecriticalpointofpbu1whichgivesrisetodx(0)intheconstructionof(pbu1,Gpcu1).Thenwedenegx(0):=u1(t0).Dene:=H(hDiP).Wenowwishtoshowthatforeachk2f1,...,Kg(andfork=0ifdx(0)isdened),gx(k)2feD.Ifk2f2,...,Kg(ork=0ifdx(0)isdened),thentk6=M,sojpu1(gx(k))j=jpu1(u1(tk))j=jv1(tk)j=j[v1(tk)j=H(h\DnE(1)iP)=H(hDiP)=.Fork=1,tk=M,sojpu1(gx(k))j=jpu1(u1(M))j=jv1(M)j=j1 1+1 2[v1(tk)j=1 1+1 2H(hcbDiP)=H(hDiP)=.Nowsupposebywayofcontradictionthatthereissomecriticalpointzofpu1ineDsuchthatjpu1(z)j>.Thenjpu1(z)j+mindiff(v1)bydenitionofmindiff.Choosesomet2f1,...,N)]TJ /F4 11.955 Tf 12.71 0 Td[(1g,suchthatu1(t)=z.Asmentionedabove,thisimpliesthatdu1(t)2bDand,sincetk=M,t6=M,sojdv1(t)j=jv1(t)j+mindiff(v1)>H(hcbDiP)byItem 3 inthechoiceof1.Thusdu1(t)isnotinoneoftheboundedfacesofcbD,which 73

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isacontradictionbythedenitionofhcbDiPC.Thereforethevaluethatjpu1jtakesateachpointinfgx(1),...,gx(K)g(andgx(0)ifitisdened)is,andthereisnocriticalpointofpu1ineDatwhichjpu1jtakesavaluelargerthan,soweconcludebyTheorem 2.1 thateachpointfx(i)isinthecriticallevelcurveineDonwhichjpu1jtakesitslargestvalue,namelyfeD.Supposenowthatdx(0)isdened.Wehavealreadyseenthatjpu1(gx(1))j=jpu1(gx(0))j.Nowforeachk2f1,...,N)]TJ /F4 11.955 Tf 12.1 0 Td[(1g,arg(dv1(k))=arg(v1(k)).ThereforebytheconstructionofhbiPC,wehavearg(pu1(gx(1)))=arg(v1(t1))=arg([v1(t1))=a(dx(1))=a(dx(0))=arg([v1(t0))=arg(v1(t0))=arg(pu1(gx(0)))Thereforepu1(gx(1))=pu1(gx(0)).Wenowwishtoshowthatgx(1)=gx(0).DeneL1tobethestraightlinepathfromgx(1)todx(1).LetLdenotetheportionofthegradientlineofpbu1whichconnectsdx(1)withdx(0).LetL0denotethestraightlinepathfromdx(0)togx(0).ByItem 1 inthechoiceof1,Item 5 inthechoiceof1,andItem 3 inthechoiceof1,itcanbeshownthatthispathfromgx(1)togx(0)doesnotintersectanycriticallevelcurveofpu1otherthanfeD.Thereforewecanprojectthispathalonggradientlinestoapath:[0,1]!feDfromgx(1)togx(0).Thenitcaneasilybeshownthateithergx(1)=gx(0),orthereissomer2(0,1)suchthat(r)isacriticalpointofpu1orarg(pu1((r)))=arg(pu1(x(1))+.HoweverbyItem 8 inthechoiceof1,Item 1 inthechoiceof2,Item 5 inthechoiceof1,andItem 3 inthechoiceof1,nosuchrexists,soweconcludethatgx(1)=gx(0).Nowchoosesomek02f1,...,Lg.Wewillnowndanedge]E(k0)offeDwhichcorrespondstotheedgeE(k0)inD. Case6.0.12. x(1)isnotanendpointofE(k0). 74

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Leti,j2f1,...,Lgbetheindicessuchthatx(i)istheinitialpointofE(k0)andx(j)isthenalpointofE(k0).RecallthatasweareviewinghbiPCasembeddedinCasthecriticallevelcurvesofpbu1,wehavecx(i)=[u1(ti)andcx(j)=[u1(tj).Anda(x(i))=a(cx(i))anda(x(j))=a(cx(j))bytheconstructionofhbiPC.Thusa(x(i))=a(cx(i))=arg(pdu1(1)([u1(ti))),andsimilarlya(x(j))=a(cx(j))=arg(pbu1([u1(tj))).Assumethata(x(i))=0(otherwisemaketheappropriateminorchangesthroughouttheargument).Let>0denotethechangeinarg(pbu1)along[E(k0),andletbbeaparameterizationof[E(k0)accordingtoarg(pbu1).Thatis,b:[0,]![E(k0),andforeachr2[0,],arg(pbu1(b(r)))=r.ByItem 3 inthechoiceof1,wemayndapath:[0,]!feDsuchthat(0)=u1(ti),andforeachr2[0,],arg(pu1((r)))=r,andj(r))]TJ /F18 11.955 Tf 12 0 Td[(b(r)j<2.Nowb()=[u1(tj),so=arg(dv1(tj))mod2.Thereforepu1(())=jv1(tj)jeiarg(v1(tj))=v1(tj).Moreover,j())]TJ /F3 11.955 Tf 11.96 0 Td[(u1(tj)jj())]TJ /F18 11.955 Tf 12 0 Td[(b()j+jb())]TJ /F7 11.955 Tf 12.86 4.13 Td[([u1(tj)j+j[u1(tj))]TJ /F3 11.955 Tf 11.95 0 Td[(u1(tj)j.However[u1(tj)=b(),sowehavej())]TJ /F3 11.955 Tf 11.95 0 Td[(u1(tj)jj())]TJ /F18 11.955 Tf 12 0 Td[(b()j+j[u1(tj))]TJ /F3 11.955 Tf 11.96 0 Td[(u1(tj)j<2+1<22.ByItem 6 inthechoiceof1andItem 1 inthechoiceof2,thereisnopointinB22(u1(tj))nfu1(tj)gatwhichpu1takesthevaluev1(tj),weconcludethat()=u1(tj).Thereforewehavethatisapathfromu1(ti)tou1(tj)throughfeD.Wenowwishtoshow 75

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thattheimageofconsistsofasingleedgeofe.Thatis,wewishtoshowthatforanyr2(0,),(r)isnotacriticalpointofpu1.Supposebywayofcontradictionthatthereissomer(0)2(0,)suchthat(r(0))isacriticalpointofpu1.Thereforewehaver0,0,2farg(v1),...,arg(vN)]TJ /F5 7.97 Tf 6.59 0 Td[(1)g,andthusbothr0and)]TJ /F3 11.955 Tf 11.99 0 Td[(r0aregreaterthandarg(1,v1(1),...,v1(N)]TJ /F5 7.97 Tf 6.59 0 Td[(1)).SincebyItem 8 inthechoiceof1andItem 1 inthechoiceof2,2
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SobyItem 2 inthechoiceof2andItem 4 inthechoiceof1,wehavejpbu1(z))]TJ /F18 11.955 Tf 15.09 4.28 Td[(dv1(l0)j<1+1+1 2<31.Thereforethepathpbu1(L)iscontainedinB31(dv1(l0)),andjdv1(l0)jminmod(v1),sobyItem 1 inthechoiceof1,weconcludethatthenetchangeinargumentofpbu1alongLislessthandarg(1,v1) 4.Asnotedabove,r(0)and)]TJ /F3 11.955 Tf 11.96 0 Td[(r(0)arebothgreaterthandarg(1,v1).Therefore,wecanchoosesomes12(0,r0),andsomes22(r0,),eachofwhichisgreaterthandarg(1,v1) 2awayfromeachmemberoffarg(v1(1)),...,arg(v1(N)]TJ /F5 7.97 Tf 6.58 0 Td[(1))g.Thereforeb(s1),b(s2)=2L.Wenowconsidertheset\DnE(1)[Lasagraphwhoseverticesaretheverticesof\DnE(1)alongwithanyintersectionsof\DnE(1)andL.(Notethatthesmoothnessoftheedgesof\DnE(1)andthesmoothnessofLimplythat\DnE(1)andLintersectatonlynitelymanyplaces.)LetK1andK2denotetheedgesof\DnE(1)[Lwhichcontainb(s1)andb(s2)respectively.SinceLintersects[E(k0)atb(r0),weconcludethatb(s1)andb(s2)areindifferentedgesof\DnE(1)[L,soK16=K2.LetG0denotetheboundedfaceof\DnE(1)suchthat[E(k0)iscontainedin@G0. Subcase6.0.12.1. BothK1andK2areadjacenttotheunboundedfaceof\DnE(1)[L.LetD1andD2denotetheboundedfacesof\DnE(1)[LwhichareadjacenttoK1andK2respectively.Theninthiscase,Lmustintersectsomeportionof@G0nb(s1,s2),andD16=D2.Bychoiceofsj,thechangeinargumentofpbu1alongKjisgreaterthandarg(1,v1) 2forj=1,2.Let)]TJ /F11 7.97 Tf 12.51 -2.45 Td[(\DnE(1)denotetheportionof@D1whichiscontainedin\DnE(1).Let)]TJ /F15 7.97 Tf 6.78 -1.79 Td[(Ldenotetheportionof@D1whichiscontainedinL.Sincetheargumentofpbu1isstrictlyincreasingon\DnE(1),andK1\DnE(1),thenetchangeintheargumentofpbu1along)]TJ /F11 7.97 Tf 12.52 -2.46 Td[(\DnE(1)isgreaterthandarg(1,v1) 2.Andsincethetotalvariationofarg(pbu1)onLislessthandarg(1,v1) 2,themagnitudeofthenetchangeinarg(pbu1)along)]TJ /F15 7.97 Tf 6.78 -1.79 Td[(Lislessthandarg(1,v1) 2.Thereforeweconcludethatthenetchangeinarg(pbu1)along@D1isgreaterthanzero.WeconcludethatD1containsazeroofpbu1.Theexactlysimilarargumentimplies 77

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thatD2containsazeroofpbu1.Letz1denoteoneofthezerosofpbu1inD1,andletz2denoteoneofthezerosofpbu1inD2.ThenTheorem 2.1 impliesthatthereisacriticallevelcurvebofpbu1inG0,suchthateachofz1andz2isinaboundedfaceofb.ThenbintersectsD1andD1c(becausez2=2sc(D1)),andthusbintersectsD1.balsointersectsD2D1c,andisconnected,thusbintersects@D1.Howeverbdoesnotintersect\DnE(1),sothereforeweconcludethatbintersectsL.Wehavealreadyshownthatforallz2L,jpbu1(z))]TJ /F18 11.955 Tf 15.4 4.28 Td[(dv1(l0)j<31.Thereforethevaluethatjpbu1jtakesonbiscontainedin()]TJ /F4 11.955 Tf 12.21 0 Td[(31,+31).BytheMaximumModulusPrinciple,sincebG0,thevaluethatjpbu1jtakesonbiscontainedin()]TJ /F4 11.955 Tf 12.29 0 Td[(31,).HoweverbytheconstructionofhbiPCandItem 9 inthechoiceof1,nocriticalvaluesofpbu1havemoduliin()]TJ /F4 11.955 Tf 12.19 0 Td[(31,),whichsuppliesuswithourcontradiction. Subcase6.0.12.2. OneofK1orK2isnotadjacenttotheunboundedfaceof\DnE(1)[L.AssumethatK1isnotadjacenttotheunboundedfaceof\DnE(1)[L.SinceK1isadjacenttotheunboundedfaceof\DnE(1),oneofthefacesof\DnE(1)[LwhichisadjacenttoK1iscontainedintheunboundedfaceof\DnE(1).LetD1denotethisface.Let)]TJ /F11 7.97 Tf 12.52 -2.46 Td[(\DnE(1)denotetheportionof@D1whichiscontainedin\DnE(1).Let)]TJ /F15 7.97 Tf 6.77 -1.79 Td[(Ldenotetheportionof@D1whichiscontainedinL.ThensinceD1iscontainedontheunboundedportionofb,arg(pbu1)isstrictlydecreasingas)]TJ /F11 7.97 Tf 12.51 -2.45 Td[(\DnE(1)istraversedwithpositiveorientation,andthechangeinarg(cpu1)asK1istraversedasaportionof@D1(thuswiththeoppositeorientationasbefore)islessthan)]TJ /F15 7.97 Tf 10.5 5.7 Td[(darg(1,v1) 2.Thereforethenetchangeinarg(pbu1)as)]TJ /F11 7.97 Tf 12.51 -2.46 Td[(\DnE(1)istraversedislessthan)]TJ /F15 7.97 Tf 10.49 5.7 Td[(darg(1,v1) 2.Andsincethetotalvariationofarg(pbu1)alongLislessthandarg(1,v1) 2,thenetchangeinarg(pbu1)along)]TJ /F15 7.97 Tf 6.77 -1.79 Td[(Lislessthandarg(1,v1) 2inmagnitude.Thereforethenetchangeinarg(pbu1)along@D1isstrictlylessthan0.Thereforethereisapoleofpbu1inD1,whichisobviouslyacontradictionaspbu1isapolynomial. 78

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Thereforebytheprevioustwosubcases,weconcludethatthereisnor2(0,)suchthat(r)isacriticalpointofpu1.Therefore(0,)isanedgeoffeD.Let]E(k0)denotethisedge. Case6.0.13. x(1)isanendpointofE(k0).Inthiscase,therearethreesubcasestoconsider,twoofwhichareessentiallyidenticaltothepreviouscase,andwhichwelistrst. Subcase6.0.13.1. k0=1(soE(k0)=@F). Subcase6.0.13.2. k06=1,butx(1)isstillavertexofDnE(1).Fortheprevioustwocases,bothendpointsof[E(k0)arecriticalpointsofpbu1,and[E(k0)iscompletelycontainedinasinglecriticallevelcurveofpbu1.(cbDforSubcase 6.0.13.1 and\DnE(1)inSubcase 6.0.13.2 .)ThereforethemethodofCase 6.0.12 maybeappliedwithminorchanges,andtheconclusionisthatifi,j2f0,...,Kgaretheindicessuchthatx(i)istheinitialvertexofE(k0)andx(j)isthenalpointofE(k0),thenthereisapathbwhichparameterizes[E(k0)withrespecttoarg(pbu1)andanedge]E(k0)offeDfromfx(i)tofx(j)whichwecall]E(k0)withaparameterizationewhichparameterizes]E(k0)withrespecttoarg(pu1)suchthatforeachr,jb(r))]TJ /F18 11.955 Tf 14.32 3.82 Td[(g(r)j<2. Subcase6.0.13.3. k06=1andx(1)isnotavertexofDnE(1).Sincek06=1,E(k0)doesnotform@F.Asnotedearlier,thefactthathcbDiPCwasformedusingthescatteringmethodimpliesthatbothendpointsofE(k0)arenotatx(1).Leti2f2,...,Kgbetheindexsothatx(i)istheotherendpointofE(k0).Assumeduringthefollowingargumentthatx(i)istheinitialpointofE(k0)(otherwisemaketheappropriateminorchanges,suchasreversingorientationsofpaths,etc.).Let2Rnf0gdenotechangeinargumentalongE(k0)fromx(i)tox(1).Assumethata(x(i))=0(otherwisemaketheappropriateminorchanges).LetEdenotetheedgeofDnE(1)whichcontainsx(1)(whichisnolongeravertexofDnE(1)).LetbEdenotetheedgeof\DnE(1)whichcorrespondstoE.Recallthat[E(k0)istheportionofbEformedinthefollowingway.Let(1)denotethechangeinargumentofpbu1alongbEbeginningat 79

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cx(i).Letd(1)beaparameterizationofbEaccordingtoarg(pbu1)andbeginningatcx(i).Sod(1):[0,(1)],andarg(pbu1(b(r)))=rforeachr2[0,(1)].Thenwedene[E(k0)tobed(1)([0,]).Letbdenotethepathd(1)restrictedto[0,].AgainbyItem 3 inthechoiceof1,thereisapath:[0,]!feDsuchthat(0)=u1(ti),foreachr2[0,]arg(pu1((r)))=randj(r))]TJ /F18 11.955 Tf 12.14 0 Td[(b(r)j1.TheargumentforthissubcaseisverysimilartotheargumentforCase 6.0.12 .Themajordifferenceisinthemethodbywhichweshowthat()=]u1(t1).Ourmethodhereissimilartothewayinwhichweshowedthatgx(0)=gx(1)(whengx(0)isdened).SincetheimageofiscontainedinfeD,weconcludethatjpu1(())j=jpu1(gx(1))j.Bydenitionof,arg(pu1(()))=.Andbydenitionof,arg(pu1(x(1)))=a(gx(1))=(sinceweareassumingthatarg(pu1(fx(i)))=0).Thereforepu1(())=pu1(x(1)).LetL1denotethestraightlinepathfromgx(1)todx(1).LetLdenotetheportionofthegradientlineofpbu1whichconnectsdx(1)tob().LetL2denotethestraightlinepathfromb()to().LetL1,L,L2denotethepathobtainedbyconcatinatingthethreepathsL1,L,andL2.ByItem 5 inthechoiceof1,Item 1 inthechoiceof2,Item 1 inthechoiceof1,andItem 3 inthechoiceof1,itcanbeshown(byconsideringthevalueofjpu1jonL1,L,L2)thatthepathL1,L,L2doesnotintersectanycriticallevelcurveofpu1otherthanfeD.Thereforewecanprojectthispathalonggradientlinestoapath:[0,1]!feDfromgx(1)to().Thenitcaneasilybeshownthateithergx(1)=(),orthereissomer2(0,1)suchthat(r)isacriticalpointofpu1orarg(pu1((r)))=arg(pu1(x(1)))+.HoweverbyItem 8 inthechoiceof1,Item 1 inthechoiceof2,Item 5 inthechoiceof1,andItem 3 ,nosuchrcanexist,soweconcludethatu1(t1)=gx(1)=().TherestoftheargumentforthissubcaseisessentiallythesameasforCase 6.0.11.2 ,soweomitit.Theconclusionwhichwedrawisasbefore.Namely,thereisapathbwhichparameterizes[E(k0)accordingtoarg(pbu1)andanedge]E(k0)offeDfromfx(i)togx(1)withaparameterizationwhichparameterizes]E(k0)accordingtoarg(pu1)suchthatforeachr, 80

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jb(r))]TJ /F6 11.955 Tf 12.14 0 Td[((r)j<2.Nowforeachk2f1,...,Lg,letg(k)bethepathwhichparameterizesgE(k).Letd(k)bethepathwhichparameterizesdE(k).WenowwishtoshowthattheorderinwhichtheedgesgE(1),...,gE(L)appeararoundfeDisthesameastheorderoftheircorrespondingedgesaroundD.NowitdoesnotquitemakesensetospeakoftheorderinwhichtheedgesdE(1),...,dE(L)appear,becausetheseedgesarenotallcontainedinasinglelevelcurveofpbu1.HoweverthereissomewaytomakesenseoftheorderofappearanceofdE(1),...,dE(L).BeginwithdE(1).Nowforeachk2f2,...,Lg,dE(k)iscontainedin\DnE(1).Selectonepointz(k)indE(k)whichisnotanendpointofdE(k).Deneforthemomenty(k)tobethepointin@bGwhichconnectstoz(k)byagradientlineofpbu1.ThenbydenitionoftheedgesdE(2),...,dE(L),theorderinwhichthepointsy(2),...,y(L)appeararound@bGisexactlyy(2),...,y(L).Therefore,bytheconstructionofhbiPC,ifwebeginatdx(1),andwetraveldownthegradientlinecontainingx(1)intobGuntilwereach\DnE(1),andbegintraversing\DnE(1),theorderinwhichwetraversetheedgesof\DnE(1)isexactlydE(2),...dE(L)..ThisisthesenseinwhichwewillsaythattheedgesdE(1),...,dE(L)appearintheorderdE(1),...,dE(L).WewillmakefurtheruseoftheprocessjustdescribedofparameterizingtheorderinwhichtheedgesofamemberofPappearbypointscontainedintheboundaryofaregionwhichcontainsthismemberofP.Letusrstdescribethisprocessmoreprecisely.LetdenoteamemberofP,andletndenotethenumberofedgese(1),...,e(n)of.ChoosesomesimpleclosedpathsuchthattheboundedfaceGofthepathcontainsinitsboundedface,andchoosendistinctpointsin@G.Foreachedgeein,drawapathfromapointinewhichisnotanendpointofetooneofthechosenpointsin@G.DothisinsuchawaythatthepathsarecontainedintheportionofGexteriortoexceptattheendpoints,sothateachedgeconnectstoadifferentpointin@G,andsothattheydonotintersect.Foreachi2f1,...,ng,lety(i)denotethepointin 81

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@Gwhichisconnectedtoe(i).Thentheorientationoftheedgese(1),...,e(n)inisthesameastheorientationofthepointsy(1),...,y(n)in@G.RecallthatbyItem 4 inthechoiceof1,thereisachoiceofpointsz(1),...,z(L)suchthatforeachk2f1,...,Lg,thefollowingholds. z(k)isingE(k)butisnotanendpointofgE(k). arg(pu1(z(k)))ismorethandarg(1,v1) 4awayfromeachoffarg(v1(1)),...,arg(v1(N)]TJ /F5 7.97 Tf 6.59 0 Td[(1))g. z(k)ismorethan21awayfromeachcriticalpointofpu1.Nowforeachk2f1,...,Lg,let(k):[0,1]!Cbeaparameterizationoftheportionofthegradientlineofpu1whichconnectsz(k)toapointin@eD.Lety(k)denotethispointin@eD.Thenrecallthat1ischosensothatforanyj,k2f1,...,Lgwithj6=k,andforanys,t2[0,1],j(j)(s))]TJ /F6 11.955 Tf 12.49 0 Td[((k)(t)j>21,andthereisnoedgeofacriticallevelcurveofpu1otherthanE(j)within21of(j)(s).Let(k)parameterizethisgradientlinesothat(k):[0,1]!Cwith(k)(0)=y(k)and(k)(1)=z(k).Denei1:=1andchoosedistinctindicesi2,...,iL2f2,...,LgsothattheorderinwhichtheedgesoffeDappeararoundfeDisgE(i1),...,gE(iL).NowbyItem 5 inthechoiceof1andItem 1 inthechoiceof2,cbDeD.Wearenowgoingtoaltereach(k)sothatitisapathfromy(k)toapointindE(k).ByItem 1 inthechoiceof2,thereisnopointinthepath(k)whichintersectsdE(l)foranyl6=k.Ifthereisanys2[0,1]suchthat(k)(s)2dE(k),thenlets0denotethesmallestsuchs.Thendened(k)tobetherestrictionof(k)to[0,s0].Ifthereisnosuchs,let(k)denotethechangeinarg(pu1)alonggE(k)andleti2f1,...,Kgdenotetheindexsuchthatfx(i)istheinitialpointofgE(k).Letr12[a(fx(i)),a(fx(i))+(k)]bechosensothatg(k)(r1)=z(k).LetL(k)denotethestraightlinepathfromg(k)(r1)tod(k)(r1).Let(k)L(k)denotethepathobtainedbyrsttraversing(k),andthentraversingL(k)fromz(k)tod(k)(r1).Then(k)L(k)isapathfromy(k)tod(k)(r1)2dE(k).Lets0bethesmallestnumberinthedomainofthepath(k)L(k)suchthat(k)L(k)(s0)2dE(k).Thendened(k)tobethispath(k)L(k)restrictedto[0,s0].NowsinceL(k)B1(z(k)),byItem 4 inthechoiceof1,d(k)doesnotintersect(l) 82

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foranyl6=k,anditcanbeshownthatL(k)doesnotintersectthegradientlinewhichconnectsx(1)to\DnE(1).Denedz(k):=d(k)(s0).Bydenitionofthe(k),theorderinwhichthepointsy(1),...,y(L)appeararound@eDbeginningwithy(1)isy(1)=y(i1),y(i2),...,y(iL).Wewishtoshowthatforeachk2f2,...,Lg,ik=k.Recallthatifonebeginsatx(k),traversesthegradientlinedownto\DnE(1),andbeginstraversing\DnE(1)withpositiveorientation,therstedgeonetraversesisdE(2).Thereforetherstpointofthesetfdz(2),...,dz(L)gwhichoneencounterswhiletraversing\DnE(1)isdz(2).Nowconsiderthepathoneobtainsbytraversingd(1)fromy(1)todz(1),traversingdE(1)withpositiveorientationfromdz(1)todx(1),traversingthegradientlineofpbu1fromdx(1)tothepointwhereitintersects\DnE(1)(letuscallthatpointzforthemoment),traversingdE(2)fromztodz(2),andnallytraversingd(2)fromdz(2)toy(2).Letdenotethispath.Sincenocz(l)isinthispathforl=2f1,2g,andno(l)forl=2f1,2gcanintersectthispath,weconcludethatifonetraverses@eDfromy(1)toy(2)withpositiveorientation,thenonedoesnotencounteranyy(l)forl=2f1,2g.Thereforesincei2istheindexsuchthaty(i2)isthenextpointin@eDaftery(1)(withrespecttoapositiveorientation),thereforeweconcludethati2=2.Thesameargumentgivesusthatforeachk2f3,...,Lg,ik=k.ThereforetheorderinwhichtheedgesgE(1),...,gE(L)appeararoundfeDisexactlygE(1),...,gE(L).ThereforetheedgesgE(k)appearinthesameorderaroundfeDasthecorrespondingedgesE(k)appeararoundD.LetGdenoteforthemomenteitherthefacebFofcbDoroneoftheboundedfacesof\DnE(1).Letn1bethenumberofedgesofbwhicharecontainedin@G.Leti1,...,in2f1,...,LgbetheindicessuchthatdE(i1),...,dE(in)aretheedgeswhichform@Glistedinorderoftheirappearancearound@G.ThengE(i1),...,gE(in)formsasimpleclosedpathand,bytheMaximumModulusPrinciple,noedgeoffeDintersectstheboundedfaceofthispath.LeteGdenotethefaceoffeDwhichhasthispathasitsboundary.Bydenitionofthepathsf(i)denedearlier,thechangeofarg(pu1)along@eGisthesameasthechangeinarg(pbu1)along@G.Thereforethenumberofzerosof 83

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pu1ineGisthesameasthenumberofzerospbu1inG.Letm2denotethenumberofboundedfacesofD.Then\DnE(1)hasm)]TJ /F4 11.955 Tf 12.39 0 Td[(1boundedfaces.DenedD(1):=bFandletdD(2),...,[D(m)beanenumerationoftheboundedfacesof\DnE(1).ThenwehavemXk=1z(dD(k))=mXk=1z(gD(k))wherez(dD(k))denotesthenumberofzerosofpbu1indD(k)andz(gD(k))denotesthenumberofzerosofpu1ingD(k).Thereforethenumberofzerosofpu1containedintheboundedfacesoffeDisgreaterthanorequaltothenumberofzerosofpbu1intheboundedfacesofcbD.HoweverbydenitionofcbDandthemap,eachzeroofpbu1whichiscontainedinbDiscontainedinoneofdD(1),...,[D(m).Moreover,bythesameargumentasaboveitmayeasilybeshownthattherearethesamenumberofzerosofpbu1inbDasthenumberofzerosofpu1ineD.ThesetwofactstogetherimplythatforeachboundedfaceeGoffeD,eG=gD(k)forsomek2f1,...,mg.ThereforefeDcontainsnoedgesotherthangE(1),...,gE(L).Wehavealreadyseenthatforeachk2f1,...,Kg,a(x(k))=a(dx(k))=arg(pbu1(dx(k)))=arg(pu1(gx(k)))=a(gx(k))and,bythedenitionofthepathsg(k)andd(k),eachgE(k)containsthesamenumberofdistinguishedpointsasdE(k),whichcontainsthesamenumberofdistinguishedpointsasE(k)byconstructionofhbiPC.WehavealsoalreadyseenthatH(hDiP)equalsjv1(tk)jforeachk2f1,...,KgandthisvalueisinturnequaltoH(hfeDiP).ThereforehDiPandhfeDiPsharealltheirauxiliarydata,andweconcludethathDiP=hfeDiP. Case6.0.14. hcbDiPCwasnotformedusingthescatteringmethod.InthiscasethegraphcbDisactuallyequaltoDasmembersofP,whichremovesmanyofthedifcultiesencounteredinCase 6.0.11 .ThereforetheargumentneededtoshowthathDiP=hfeDiPandestablishthecorrespondencebetweenthevertices,edges,anddistinguishedpointsofhDiPandhfeDiPisamuchsimpliedversionofthatfoundinCase 6.0.11 ,soweomitithere. 84

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NotethatCase 6.0.11 andCase 6.0.14 assumethathDiPisagraphmemberofP.IfhDiPisasinglepointmemberofPitiseasytoshowthathfeDiPmustbethesamesinglepointmemberofPbysimplyconsideringthedifferentvaluesthatjv1(k)jcantake,andusingthefactthatbytheconstructionofhbiPC,z(D)=z(bD),and,asdescribedabove,z(bD)=z(eD).Wenowwishtoshowthatthecorrespondencewehaveestablishedbetweenthegraphs,vertices,anddistinguishedpointsofhiPCandthegraphs,vertices,anddistinguishedpointsofheiPCrespectsthegradientmapsofhiPCandheiPC.Asbefore,lethiPCbeamemberofPCusedtoconstructhiPC.LetDbeaboundedfaceof,andlethbiPC,bD,andhcbDiPCandheiPC,eD,andhfeDiPCbetheobjectsforhbiPCandheiPCwhichcorrespondtohiPC,D,andhDiPCrespectively.ChoosesomeedgeE1of@DandletcE1andfE1bethecorrespondingedgesin@bDand@eD.Lety1andy2betheinitialandnalpointsofE1,andletby1,by2,ey1,andey2bethecorrespondingpointsforhbiPCandheiPC.Let1denotethechangeinargumentalongE1.Assumethata(y1)=0,(otherwisemaketheappropriateminorchanges).Thenletd(1):[0,1]!CbethepathwhichparameterizescE1accordingtoarg(pbu1).Letg(1):[0,1]!CbethepathwhichparameterizesfE1accordingtoarg(peu1).LetybeadistinguishedpointinE1.LetbyandeybethecorrespondingpointsincE1andfE1.Thenbychoiceofd(1)andg(1),jby)]TJ /F18 11.955 Tf 12.23 .5 Td[(eyj<2.DeneztobethedistinguishedpointinDsuchthatgD(y)=z.LetbzandezbethedistinguishedpointscorrespondingtozforhbiPCandheiPC.ThensincegD(y)=z,thegoalistoshowthatgeD(ey)=ez.LetE2denoteoneoftheedgesofDwhichcontainsz(ifzisavertexofDthenitwillbecontainedinmorethanoneedgeofD).Letz1andz2betheinitialandnalpointsofE2.Let2denotethechangeinargumentalongE2.Letd(2),g(2):[a(z1),a(z1)+2]!CbethepathswhichparameterizecE2andfE2withrespecttoarg(pbu1)andarg(pu1)respectively.Thenbychoiceofd(2)andg(2),jbz)]TJ /F18 11.955 Tf 12.11 .5 Td[(ezj<2.Wewillshowthedesiredresultinthecase 85

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wherehcbDiPCwasformedusingthescatteringmethod.Asbefore,theothercasesarejustsimplerversionsofthiscase. Case6.0.15. hcbDiPCwasformedusingthescatteringmethod.RecallthatbFdenotesthefaceofcbDtowhichhFiPCwasassigned,andbGdenotestheotherfaceofcbD.Wenowwilldeneapathbfrombytobz. Subcase6.0.15.1. z2@F.Inthiscasebzisadistinguishedpointin@bF.Bydenitionofbyandbz,gbD(by)=bz.Thereforethereisaportionofagradientlineb:[0,1]!Cofpbu1whichconnectsbyandbzandsuchthatb((0,1))iscontainedintheportionofbDwhichisexteriortocbD. Subcase6.0.15.2. z=2@F.Inthiscasebythedenitionofthecorrespondencealreadyestablished,bzisapointinanedgeof\DnE(1).Recallthath\DnE(1)iPChasbeenassignedtobGduringtheconstructionofhcbDiPC,andbythisconstruction,gbD(by)isapointin@bG,andonecanshowthatgbG(gbD(by))=bz.Thereforethereisaportionofagradientlineb1:[0,1]!Cofpbu1whichconnectsbytogbD(by),andanotherportionofagradientlineb2:[0,1]!Cofpbu1whichconnectsgbD(by)tobz.Letbdenotetheconcatenationofthesetwopaths.Thereforewehavethedesiredpathb.ByItem 3 inthechoiceof2andItem 2 inthechoiceof1,weconcludethatthereisapath:[0,1]!Csuchthat(0)=eyand(1)=ezand,forallr2[0,1],arg(pu((r)))=0andj(r))]TJ /F18 11.955 Tf 12.85 0 Td[(b(r)j<1.Moreover,sincejpbu1jisstrictlydecreasingonb,wemayassumethatjpu1jisstrictlydecreasingon.Thereforeforeachr2(0,1),jpu1((r))j2(jpu1(ez)j,jpu1(ey)j).Thereforeforeachr2(0,1),(r)isintheportionofeDwhichisintheunboundedfaceoffeD.ThereforebydenitionofgeD,weconcludethatgeD(ey)=ez.Thereforethecorrespondenceestablishedabovebetweenthegraphs,vertices,anddistinguishedpointsofhiPCandthoseofheiPCrespectsthegradientmapsofhiPCandheiPC.FinallyweconcludethathiPCandheiPCshareallauxiliarydata,andarethusequal. 86

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Byinspectingthisproofweimmediatelyhavethefollowingcorollary. Corollary4. Forany(f,G)2Hathereisapolynomial(p,Gp)suchthat(p,Gp)(f,G). 87

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APPENDIXASEVERALRESULTS TheoremA.1. Letpbeapolynomial.Thecriticalpointsofparecontainedintheconvexhullofthezerosofp. Proof. Letw1,w2,...,wn2Cbethezerosofprepeatedaccordingtomultiplicity.Assumebywayofcontradictionthatthereissomecriticalpointz0ofpwhichisnotintheconvexhulloffwi:1ing.Bypre-composingwithalinearmap,wemayassumethatallzerosofparecontainedinthediskD=fz2C:jzj<1g,andthatz02(1,1).LetGdenoteoneoftheboundedfacesofz0whichisincidenttoz0,andlet:[0,1]!CbeaparameterizationoftheboundaryofG.Thus,isasimpleclosedpathwith(0)=(1)=z0.TheMaximumModulusTheoremimpliesthatGcontainsazeroofp,andtherefore@GintersectsthelineL:=fz2C:Re(z)=1g.Dener1,r22(0,1)byr1=min(r2[0,1]:(r)2L)andr2=max(r2[0,1]:(r)2L).Then(0,r1)and(r2,1)arepathsfromz0toLwhichdonotintersectexceptatz0(andpossiblyinL).Thereforesincethereareatleasttwoboundedfacesofz0whichareincidenttoz0,thereareatleastfourpathsinz0fromz0toLwhichdonotintersectexceptatz0andinL.Itisnothardtoseethenthatthereissomes2()]TJ /F4 11.955 Tf 9.3 0 Td[(1,1)nf0gsuchthattwoofthepathsintersectthesetfz2C:Re(z)>1,Im(z)=sg.Letz1,z2bedistinctpointsinfz2C:Re(z)>1,Im(z)=sgwhicharecontainedinz0,withRe(z1)
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Denition40. For(f,G)afunctionelement,andKG,thenwedeneK:=[z2Kz. Corollary5. Let(f,G)beaspecialtypefunctionelement.LetKGbecompact.ThenKiscompact. Proof. SinceGisbounded,Kisbounded,andthusitsufcestoshowthatKisclosedinC.Letz02Kcbegiven.Wewillshowthatthereissome>0suchthatB(z0)Kc. CaseA.1.1. z02cl(G).Sincez0=2K,z0\K=;.ThusbyProposition 2.4 ,thereissomenon-criticallevelcurveLoffinGsuchthatz0iscontainedinonefaceofLandKiscontainedintheotherfaceofz0.LetD1denotethefaceofLwhichcontainsz0andletD2denotethefaceofLwhichcontainsK.Foreachz2K,z2D2andzisconnectedanddoesnotintersectL,soziscontainedinD2.ThereforeKD2.Ifwechoose>0smallenoughthatB(z0)D1,thenB(z0)Kc. CaseA.1.2. z0=2cl(G).Sincez0=2cl(G),thereissome>0suchthatB(z0)Gc,andthusB(z0)Kc.WeconcludethatKcisopeninC,andthusKisclosedinC. Denition41. If(f,G)isaspecialtypefunctionelement,andisalevelcurveoffinG,andDisafaceof,thenwesaythatfisincreasingintoDifthereissome>0suchthatjfj>onfz2D:d(fzg,)0suchthatjfj
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pointinD.Aisclosed,sobyProposition 2.4 ,thereissomenon-criticallevelcurveoffcontainedinD\GwhichseparatesfromA.CallthislevelcurveL.Let>0besuchthatjfjonL.SinceLseparatesAfrom,LdoesnotintersectA,so6=. CaseA.1.3. >.LetD1denotethefaceofLwhichcontains,andletD2denotethefaceofLwhichcontainsA.DeneD0:=D1\D,theportionofDwhichisbetweenLand.DnD1=cl(D2),soDnD1iscompactandsimplyconnected.Moreover,wemayuseLemma 1 toshowthatDncl(D2)=D\D1isconnected.Thereforesincejfj6=onD\D1,andjfjiscontinuous,eitherjfj>onD\D1,orjfjonD\D1.Dene:=d(L,).Ifw2fz2D:d(fzg,).ThusfisincreasingintoD. CaseA.1.4. <.Thesameargumentasaboveworks,withtheconclusionthatfisdecreasingintoD.IfDistheunboundedfaceof,thesameargumentworkswiththeappropriateminorchanges. Denition42. ForX,YC,dened1(X,Y):=supx2X(d(fxg,Y)),anddened2(X,Y):=supy2Y(d(X,fyg)).ThenweletddenotetheHausdorffmetricd(X,Y):=max(d1(X,Y),d2(X,Y)).IfeitherXorYareempty,wedened(X,Y):=1. PropositionA.1. Let>0begiven.Thenthereissome2(0,)suchthatforeach2()]TJ /F6 11.955 Tf 12.28 0 Td[(,+),thereissomecollectionL1,...,LNoflevelcurvesoffcontainedinGsuchthatjfjoneachLi,andd N[i=1Li!,!<. 90

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Proof. Let>0begiven.LetE1,E2,...,ENbeanenumerationoftheedgesof,andforeachi2f1,2,...,Ng,letzibesomexedinteriorpointofEi(thatis,apointinEiwhichisnotanendpointofEi).Ourrstgoalistondan>0smallenoughsothateachofthefollowinghold: 1. Foreachi2f1,2,...,Ng,theonlyfacesofthatintersectB(zi)arethetwofacesadjacenttoEi. 2. IfDisoneofthefacesof,andz2Dislessthanawayfrom,thend(z,@D)< 2. 3. IfDisoneofthefacesof,andwedeneA:=fz2D:d(fzg,)onA.Supposeanmaybefoundwhichsatiseseachofthesethreeitems.Ournextgoalistondsome2(0,)smallenoughsothatif2()]TJ /F6 11.955 Tf 12.47 0 Td[(,+),thenforeachi2f1,2,...,Ng,thereissomepointinB(zi)atwhichjfjtakesthevalue.Supposesuchan>0maybefound.Wenowshowthatthestatementofthepropositionholdsforthisand.Let2()]TJ /F6 11.955 Tf 12.06 0 Td[(,+)begiven.Ofcourseif=,thenbyputtingN=1andL1=,thestatementofthepropositionobviouslyholds,soletusassumethat6=.Foreachi2f1,2,...,Ng,letwibeaxedpointinB(zi)atwhichjfjtakesthevalue,whichmaybefoundbyItem 3 above.DeneL:=N[i=1wi. ClaimA.1.1. d(L,)<.Letx2Lbegiven.Leti2f1,2,...,Ngbesuchthatx2wi.Sinced(fwig,)<,wehavethatd(wi,)< 2byItem 2 ,andthusd(fxg,)< 2.Therefored1(L,)<.Letx2begiven.Leti2f1,2,...,Ngbesuchthatx2Ei.LetDdenotethefaceofwhichcontainswi.d(fzig,fwig)<,sobyItem 1 above,Ei@D.Andd(fwig,)<,sobyItem 2 above,d(wi,@D)< 2.Sincex2Ei@D,weconcludethatd(wi,fxg)< 2.Sincex2waschosenarbitrarily,weconcludethatd2(L,)<. 91

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Finallyweconcludethatd(L,)<,andwearedone,subjecttondingthespeciedand.Ofcourseifweshowthatapositiveconstantmaybechosentosatisfyeachofthethreeaboveitemsindividually,thentheminimumofthethreeconstantswillhavethepropertiesdesiredinachoiceof.WerstshowthatmaybechosentosatisfyItem 1 .Leti2f1,2,...,Ngbegiven.DeneEi0tobethepointsinEiwhicharenotendpointsofEi.ThennEi0isclosed,sori:=d(fzig,nEi0)>0.NowifDisafaceofsuchthatzi=2@D(andthus@DnEi0),andw2D,thenanypathfromwtoziintersects@D,inparticularthestraightlinepath.Therefored(fzig,fwg)d(fzig,@D)d(fzig,nEi0)=ri,andthereforeweconcludethatDdoesnotintersectBri(zi).Ifwechoose>0smallerthaneachri,thissatisesItem 1 .WenowshowthatmaybechosentosatisfyItem 2 .LetDbeoneofthefacesof.Bythecompactnessof@D,thereisanitesequenceofpointsx1,x2,...,xk2Dsuchthatd k[i=1fxig,@D!< 4.DeneK:=fxigki=1[z2D:d(fzg,@D) 4.ByProposition 2.4 ,thereissomeD>0suchthatifz2DislessthanDawayfrom,thenzcontainsinonefaceandKintheother.Fixsomew2DlessthanDawayfrom. ClaimA.1.2. d(w,@D)< 2.Letw02wbegiven.SincewiscontainedinDnK,wehavewz2D:d(fzg,@D)< 4. 92

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Thusd(fw0g,@D)< 4,andthusd1(w,@D)< 2.Letz02@Dbegiven.Thenforsomei2f1,2,...,kg,d(fxig,fz0g)< 4.Butxi2K,soxiandz0areindifferentfacesofw.Sincexiandz0areindifferentfacesofw,thestraightlinepathfromz0toxiintersectsw,andthusd(w,fz0g)d(fxig,fz0g)< 4.Thusd2(w,@D)< 2.Thereforeweconcludethatd(w,@D)< 2.Dene:=min(D:Disafaceof).ThenthissatisesItem 2 .FinallythefactthatmaybechosentosatisfyItem 3 followsdirectlyfromCorollary 6 .Thefactthatan>0maybefoundwiththedesiredpropertyfollowsdirectlyfromtheOpenMappingTheorem.Sincethedesired>0and>0maybefound,wearedone. 93

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FigureA-1. Gauss'Theorem 94

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APPENDIXBSEVERALLEMMATA Lemma1. ForanydisjointclosedsetsX,Ycl(C),andx,y2cl(C)n(X[Y),ifxandyareinthesamecomponentofXcandthesamecomponentofYc,thenxandyareinthesamecomponentof(X[Y)c. Proof. Supposebywayofcontradictionthatxandyareindifferentcomponentsof(X[Y)c.Assumewithoutlossofgeneralitythaty=1.LetA1denotethecomponentof(X[Y)cwhichcontainsx,andletB1denotethecomponentof(X[Y)cwhichcontainsy.LetZdenotetheunionofallboundedcomponentsofA1c.DeneA2:=A1[Z.SinceB1isopenandcontains1,andA2doesnotintersectB1,wemayconcludethatA2isbounded.ThereforeA2hasonlyasingleunboundedcomponent,soA2issimplyconnected.Andtheboundaryofasimplyconnectedsetisconnected,so@A2X[Yisconnected.BecauseXandYaredisjointandcompact,thisimpliesthat@A2iseithercontainedinXorcontainedinY.HoweverthisisacontradictionbecausexandyareinthesamecomponentofXcandinthesamecomponentofYc. ItmayeasilybeseenthatthislemmaimpliesthatifAisanopenconnectedset,andXAiscompact,suchthatXchasasinglecomponent,thenAnXisconnected.Thisfactwillbeusedseveraltimesinthispaper,andwewillcitetheabovelemmawhenitisneeded. Lemma2. Letbeaniteconnectedgraphembeddedintheplane.Ifhasthepropertythateachedgeofisintheboundarybothofaboundedandtheunboundedfaceof,someboundedfaceofhasasingleedgeofasitsboundary. Proof. WebeginbyconstructingagraphTfrom.Twillhavetwokindsovvertices.WeplaceaV-typevertexforTateachvertexof,andweplaceoneF-typevertexforTineachboundedfaceof.LetubeanF-typevertexofT.LetDdenotetheboundedfaceofwhichcontainsu.ThenwedrawanedgefromutoeachV-typevertexofTwhicharisesfromavertexofwhichiscontainedin@D.Wedrawtheseedgesin 95

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suchawaythattheyarecontainedinD(exceptattheendpoints)anddonotintersect(exceptatu).HavingdonethisforeachFtypevertexinT,theresultingconnectedgraphisT.WenowwishtoshowthatTisatree.SupposebywayofcontradictionthatTcontainssomecycleC:u1E1u2unEnu1.ConsiderthiscycleasapathinC.Reducethiscycleifnecessarysothatitformsasimpleclosedpath.LetD1denotethefaceofwhichcontainsu1.SinceCisasimplepath,theonlyedgesinCwhichhaveu1asanendpointareE1andEn.ThereforeCbisectsD1.LetEbeoneoftheedgesofwhichisin@D1andwhichiscontainedintheboundedfaceofthepathC.ThensinceTiscontainedintheclosureoftheboundedfacesof,Eisnotadjacenttotheunboundedfaceof,whichisacontradictionofthedenitionof.NowthatwehaveshownthatTisatree,andTiscertainlynite,letudenoteoneoftheleavesofT.Sinceeachvertexofisincidenttomorethanoneboundedfaceof,umustbeanF-typevertexofT.LetDnowdenotethisfaceof.SinceuisaleafofT,@Donlycontainsonevertexof,andthus@Dconsistsofasingleedgeof. Lemma3. Givenanyspecialtypefunctionelement(f,G)and>0,andanycompactsetG0Gwhichdoesnotcontainanycriticalpointsoff,thereexists>0suchthatifgisanalyticonG,andjf(z))]TJ /F3 11.955 Tf 11.95 0 Td[(g(z)j
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soifwedene(2):=inf(h(2)(z):z2G0),wemayconcludethat(2)>0.Nowdene:=(2) 100.LetgbeanalyticonG,withjf)]TJ /F3 11.955 Tf 11.05 0 Td[(gj0begiven.Thereisa(1)>0suchthatthefollowingholds.Letv2Kbechosen,suchthatdarg(v)>(1).Letu2)]TJ /F5 7.97 Tf 6.59 0 Td[(1(v)bechosen.Thenifrr(1),andisanycomponentofEpu,r,andEisanyedgein,thenthereissomepointzinEwhichisgreaterthan(1)awayfromeachcriticalpointofpu. 97

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Proof. Bydenitionofpu,therearepolynomialsP(1),...,P(n)]TJ /F5 7.97 Tf 6.58 0 Td[(1)inn)]TJ /F4 11.955 Tf 12.34 0 Td[(1variablessuchthatforu2Cn)]TJ /F5 7.97 Tf 6.58 0 Td[(1,pu(z)=1 nzn+n)]TJ /F5 7.97 Tf 6.59 0 Td[(1Xk=1P(k)(u)zk.Thereforeforz2C, jpu(z)j1 njzjn)]TJ /F18 11.955 Tf 11.95 20.45 Td[( n)]TJ /F5 7.97 Tf 6.59 0 Td[(1Xi=kjP(k)(u)jjzjk!.(B)Sinceisproper(by[ 14 ]),)]TJ /F5 7.97 Tf 6.59 0 Td[(1(K)iscompact.ByinspectingEquation B ,weconcludethatthereissomeconstantS>0suchthatifu2)]TJ /F5 7.97 Tf 6.59 0 Td[(1(K),andjzj>S,thenjpu(z)j2.ThereforeGpuBS(0)foreachu2)]TJ /F5 7.97 Tf 6.58 0 Td[(1(K).IncreaseSfurtherifnecessarysothat)]TJ /F5 7.97 Tf 6.58 0 Td[(1(K)BS(0).FinallywesetT:=sup(jpu0(z)j:u2BS(0)andz2BS(0)).Byasimilarargumentasabove,thisTisnitebythecompactnessofthesetsinvolved.Wenowdene(1):=r(1)sin((1) 2) T.Nowchooseanyv2Ksuchthatdarg(v)>(1),andanyu2)]TJ /F5 7.97 Tf 6.58 0 Td[(1(v).Letr2[r(1),1)bechosen,andletbesomecomponentEpu,r,andletEbeanyedgeof.SincetheendpointsofEarecriticalpointsofpu,thechangeinargumentofpualongEisgreaterthanorequalto(1).Thereforethereissomepointz(1)inEsuchthatdarg(pu(z),v(k))>(1) 2foreachk2f1,...,n)]TJ /F4 11.955 Tf 12.15 0 Td[(1g.Fixsomei2f1,...,n)]TJ /F4 11.955 Tf 12.15 0 Td[(1g.Theanglebetweenpu(z(1))andpu(u(i))isgreaterthanorequalto(1) 2,andjpu(z(1))j=rr(1).Thereforebygeometry,jpu(z(1)))]TJ /F3 11.955 Tf 12.23 0 Td[(pu(u(i))jr(1)sin((1) 2).LetLdenotethestraightlinepathfromz(1)tou(i).Thenjpu(z(1)))]TJ /F3 11.955 Tf 11.95 0 Td[(pu(u(i))j jz(1))]TJ /F3 11.955 Tf 11.95 0 Td[(u(i)jmax(jpu0(z)j:z2L)T.Therefore 98

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jz(1))]TJ /F3 11.955 Tf 11.96 0 Td[(u(i)jjpu(z(1)))]TJ /F3 11.955 Tf 11.95 0 Td[(pu(u(i))j Tr(1)sin((1) 2) T=(1).Sincethisholdsforeachi2f1,...,n)]TJ /F4 11.955 Tf 11.95 0 Td[(1g,wearedone. Lemma5. Letv2Cn)]TJ /F5 7.97 Tf 6.58 0 Td[(1and>0begiven.Thenthereexistsa>0suchthatifbv2Cn)]TJ /F5 7.97 Tf 6.59 0 Td[(1andjv)]TJ /F18 11.955 Tf 12.97 .5 Td[(bvj<,andbu2)]TJ /F5 7.97 Tf 6.58 0 Td[(1(bv),thenthereisau2)]TJ /F5 7.97 Tf 6.58 0 Td[(1(v)suchthatju)]TJ /F18 11.955 Tf 12.02 .5 Td[(buj<. Proof. Itwasshownin[ 14 ]thatiscontinuous,open,andproper,andthat)]TJ /F5 7.97 Tf 6.59 0 Td[(1(v)isnite.Sinceisopenand)]TJ /F5 7.97 Tf 6.59 0 Td[(1(v)isnite,thereissome>0smallenoughthatB(v)\u2)]TJ /F13 5.978 Tf 5.76 0 Td[(1(v)(B(u)).Sinceisproper,)]TJ /F5 7.97 Tf 6.59 0 Td[(1(cl(B(v)))iscompact.Supposebywayofcontradictionthatthereisasequenceoffvkg1k=0B(v)suchthatvk!v,andforeachk0thereisauk2)]TJ /F5 7.97 Tf 6.58 0 Td[(1(vk)suchthatjuk)]TJ /F3 11.955 Tf 12.66 0 Td[(uj>foreachu2)]TJ /F5 7.97 Tf 6.58 0 Td[(1(v).Since)]TJ /F5 7.97 Tf 6.58 0 Td[(1(cl(B(v)))iscompact,thereissomesub-sequencefuklg1l=0whichconvergestosomepointu.Sinceiscontinuous,(u)=v,sou2)]TJ /F5 7.97 Tf 6.58 0 Td[(1(v),whichgivesusourcontradiction. Denition43. If:[,]!Cisapath,andfisafunctionwhichisanalyticandnon-zeroontheimageof,thenwesaythatisparameterizedaccordingtoarg(f)ifforeachr2[,],arg(f((r)))=r. Lemma6. Letv2Vn)]TJ /F5 7.97 Tf 6.59 0 Td[(1,and(1)>0begiven.Thereexistssome(2)2(0,(1))suchthatifu2)]TJ /F5 7.97 Tf 6.59 0 Td[(1(v),andisacriticallevelcurveof(pu,Gpu)(withjfj>0on),andx2,thenify2B(2)(x)satisesjf(y)j=,thenthereisapathfromytoxwhichiscontainedin\B(1)(x).Moreover,wemaychoosesothatarg(pu)isstrictlyincreasingorstrictlydecreasingalong,andparameterizedaccordingtoarg(pu). Proof. Since)]TJ /F5 7.97 Tf 6.58 0 Td[(1(v)isnite([ 14 ]),weneedonlyshowtheresultforsomexedu2)]TJ /F5 7.97 Tf 6.59 0 Td[(1(v).Letu2)]TJ /F5 7.97 Tf 6.59 0 Td[(1(v),andletbeoneofthecriticallevelcurvesof(pu,Gpu),(withjfj>0on).Letx2begiven.Letk2Ndenotethemultiplicityofxasazeroofpu0(possiblyk=0).ThenthereissomeneighborhoodDB(1)(x)ofxandS>0 99

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andconformalmap:D!BS(pu(0))suchthatpu(z)=(z)k+1+pu(x)forallz2D.Denef(w):=wk+1+pu(x).Thelevelcurvesoffarewellunderstood.LetLdenotethelevelcurveoffwhichcontains0.Thenifw2L,thereisapathinLfromwto0whichiscontainedinBjwj(0),alongwhicharg(f)iseitherstrictlyincreasingorstrictlydecreasing.Choosesomer>0suchthatBr(x)D.Lety2Br(x)beanypointsuchthatjpu(y)j=.Let(1)denotethepathinBj(y)j(0)from(y)to0alongwhicharg(f)isstrictlyincreasingorstrictlydecreasing.Thenifwedene:=)]TJ /F5 7.97 Tf 6.59 0 Td[(1(1),)]TJ /F5 7.97 Tf 6.58 0 Td[(1(Bj(y)j(0))DB(1)(x),andforeacht2[0,1],pu((t))=f((1)(t)),soarg(pu)iseitherstrictlyincreasingorstrictlydecreasingalong.Nowforx2,leth(x)denotethesupremumoverallr>0suchthatfory2Br(x)withjpu(y)j=,apathwiththedesiredpropertiesmaybefound.Wehavejustshownthath(x)>0forallx2,anditiseasytoseethathiscontinuous,soifwedeneh():=inf(h(x):x2),thecompactnessofimpliesthath()>0.Thenchoosing(2)>0tobelessthanh()foreachcriticallevelcurveofpuonwhichjpuj6=0,itisclearthat(2)hasthedesiredproperty. Lemma7. Letv2Vn)]TJ /F5 7.97 Tf 6.59 0 Td[(1,and(1)>0begiven.Thereexistssome(2)2(0,(1))suchthatifu2)]TJ /F5 7.97 Tf 6.58 0 Td[(1(v),andisacriticallevelcurveof(pu,Gpu)(withjpuj>0on),andx2,thenify2B(2)(x)satisesarg(pu(y))=arg(pu(x)),thenthereisapathfromytoxwhichiscontainedinB(1)(x)andsuchthatarg(pu((r)))=arg(pu(x))forallr.Moreoverwemaychoosesothatjpujisstrictlyincreasingorstrictlydecreasingalong,andparameterizedaccordingtojpuj. Proof. EssentiallythesameargumentforLemma 6 workshere. Lemma8. Letv2Vn)]TJ /F5 7.97 Tf 6.58 0 Td[(1,and>0andbegiven.Thenthereexistsa>0suchthatifu2)]TJ /F5 7.97 Tf 6.58 0 Td[(1(v),andbu2)]TJ /F5 7.97 Tf 6.58 0 Td[(1(Vn)]TJ /F5 7.97 Tf 6.59 0 Td[(1)suchthatju)]TJ /F18 11.955 Tf 12.47 .5 Td[(buj<,thenthefollowingholds.Gpbu,1Gpu,2,andjpu(z))]TJ /F3 11.955 Tf 11.96 0 Td[(pbu(z0)j
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Proof. Thisfollowsfromthefactthatthecoefcientsofpuarepolynomialsinthecomponentsofu.Thereforeifu(n)!uin)]TJ /F5 7.97 Tf 6.58 0 Td[(1(Vn)]TJ /F5 7.97 Tf 6.59 0 Td[(1),thenpu(n)!puuniformlyonanycompactset. Denition44. Foru2Cn)]TJ /F5 7.97 Tf 6.59 0 Td[(1,if:[0,1]!Cisapath,and00suchthatforallr2(a)]TJ /F6 11.955 Tf 12.11 0 Td[(,a)[(b,b+),(r)islessthan(1)awayfromanycriticalpointofpu. Foreachr2(a,b),(r)isgreaterthanorequalto(1)awayfromeverycriticalpointofpu. Thereissomer2(a,b)suchthat(r)isgreaterthan(2)awayfromeverycriticalpointofpu. Denition45. Letu2Cn)]TJ /F5 7.97 Tf 6.59 0 Td[(1begiven.LetbeapathinEpu,forsome>0,suchthat(0)and(1)arecriticalpointsofpu.For0a0.Thenthereexistsaconstant>0suchthatthefollowinghold.Letu2)]TJ /F5 7.97 Tf 6.59 0 Td[(1(v)bechosen,andxsomebu2B(u)suchthatifwedenebv=(dv(1),...,\v(n)]TJ /F5 7.97 Tf 6.59 0 Td[(1)):=(bu),thenarg(dv(k))=arg(v(k))foreachk2f1,...,n)]TJ /F4 11.955 Tf 12.78 0 Td[(1g.Forsomek2f1,...,n)]TJ /F4 11.955 Tf 12.78 0 Td[(1gwithjv(k)j6=0,letbdenotethelevelcurveofpbuwhichcontainsdu(k).LetbEdenotesomeedgeofbwhichisincidenttodu(k),andletbdenoteaparameterizationofbEsuchthatb:[,]!b(forsome,2R)satisesb()=du(k)andarg(pbu(b(t)))=tforallt2[,].(Note:ifarg(pbu)isincreasingasthisportionofbEistraversed,then<,otherwise>.)Thenifweletdenotethecriticallevelcurveofpucontainingu(k),thereisapath:[,]!suchthat()=u(k),andforeacht2[,],arg(pu((t)))=tandj(t))]TJ /F18 11.955 Tf 11.99 0 Td[(b(t)j<(1). 101

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Proof. Weassumethatarg(pbu)isincreasingasbistraversed.Otherwisemaketheappropriatechanges.Wewillshowthattheresultofthelemmaholdsforanyxedu2)]TJ /F5 7.97 Tf 6.59 0 Td[(1(v),whichwillsufcebecause)]TJ /F5 7.97 Tf 6.59 0 Td[(1(v)isniteby[ 14 ].Reduce(1)>0ifnecessarysothatforeachk2f1,...,n)]TJ /F4 11.955 Tf 12.55 0 Td[(1gwithjv(k)j6=0,ifjz)]TJ /F3 11.955 Tf 12.55 0 Td[(u(k)j<(1),thenjpu(z))]TJ /F3 11.955 Tf 10.71 0 Td[(v(k)j0smallenoughsothatforeachk2f1,...,n)]TJ /F4 11.955 Tf 12.23 0 Td[(1g,B2M(v(k))pu(B(2)(u(k))).ByLemma 8 ,wemaychoosea(1)>0sothat(1)<(2) 2,andifbu2B(1)(u),thenjpu(z))]TJ /F3 11.955 Tf 12.18 0 Td[(pbu(bz)j0suchthatforeachx2Kthefollowingholds.Letl2f1,...,n)]TJ /F4 11.955 Tf 11.95 0 Td[(1gbechosensothatjpu(x)j=jv(l)j. foreachk2f1,...,n)]TJ /F4 11.955 Tf 11.96 0 Td[(1g.DeneG0:=fx2Gpu:d(x,@Gpu),d(x,u(k))foreachkg.ByLemma 3 ,wemaychoose>0sothat
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Foranyw2B(pu(x)),thereisay2B(x)withf(y)=w. Foranyw2B(f(x)),thereisay2B(x)withpu(y)=w.ByLemma 8 andthecontinuityof,wemaychoose2(0,(1))sothatifbu2B(u),thenjpu(z))]TJ /F3 11.955 Tf 12.7 0 Td[(pbu(bz)jminmod(v) 4,andjv)]TJ /F18 11.955 Tf 12.89 .5 Td[(bvj< 4,wehavejdv(k)j>minmod(v) 2.Letbdenotethelevelcurveofpbuwhichcontainsdu(k).LetbEdenotesomeedgeofbwhichisincidenttob.Letdenotesomechoiceoftheargumentofpbu(du(k)),andletb:[,]!b(forsome>becausearg(pbu)isincreasingasbistraversed)beapathwhichparameterizesbEaccordingtotheargumentofpbu(thatis,arg(pbu(b(t)))=tforallt2[,]).Notethatbythedenitionofa((1),(1))tripoveraninterval,ifbtakes((1),(1))tripsovertwointervalsI(1),I(2)[0,1],theneitherI(1)=I(2),orI(1)andI(2)aredisjoint.Thereforesincebisarectiablepath,btakesatmostnitelymanydistinct((1),(1))trips. CaseB.0.5. btakesa[(1),(1)]triponsomesub-intervalof[,].Let[r(1),s(1)],...,[r(N),s(N)][,]bethedisjointsubintervalsof[,]overwhichtakes((1),(1))trips,orderedsothats(k)
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SincepuisinjectiveonB(b(r))foreachr2[r(j),s(j)],andpuisanopenmapping,itiseasytoshowthatisacontinuousfunction,andthusapathfrom(r(j))to(s(j)).Further,ifr2[r(j),s(j)],jpu((r))j=jw(r)j=jv(k)j.Thereforeweconcludethatj[r(j),s(j)]isapathinEpu,jv(k)j,andbyconstruction,foreachr2[r(j),s(j)],jb(r))]TJ /F6 11.955 Tf 12.47 0 Td[((r)j
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decreasingon(1),thensincepuisinjectiveonD,foreachr2(s(j),t(0)),(1)(r)=(r).Furthermore,sincepuisinjectiveinaneighborhoodofeachpointof([r(j),s(j)]),(1)mustcontinuetotracebackalongtheentirelengthof([r(j),s(j)]).Thisisbecauseboth(1)andareparameterizedaccordingtoarg(pu),soanybranchingoffof(1)fromwouldhavetobeacriticalpointofpu.However(1)maynottracebackalong([r(j),s(j)])becausetheimageof(1)iscontainedinB(1)(u(l)).Thereforeweconcludethatarg(pu)isincreasingon(1).Byverysimilarreasoningwemayobtainapath(2)fromu(l)to(r(j+1))containedinB(1)(u(l))\Epu,jv(l)jparameterizedaccordingtoarg(pu),andalongwhicharg(pu)isincreasing.Lets(j)bethechoiceofarg(pu((s(j))))whichisthestartingpointforthedomainof(1),andchoosesomet(1)>0sothatthedomainof(1)is[s(j),s(j)+t(1)].Nowlets(j)+t(1)bethechoiceofarg(v(l))whichisthestartingpointforthedomainof(2),andchooset(2)>0sothatthedomainof(2)is[s(j)+t(1),s(j)+t(1)+t(2)].Let:[s(j),s(j)+t(1)+t(2)]!B(1)(u(l))denotetheconcatenationof(1)and(2).Thens(j)+t(1)+t(2)=r(j+1)(mod2),sot(1)+t(2)=r(j+1))]TJ /F3 11.955 Tf 12.42 0 Td[(s(j)(mod2).Howeverbychoiceof(1),thetotalchangeinargumentofpualongmustbelessthan.Andsinceb(s(j),r(j+1))B(1)(u(l)),thetotalchangeinargumentofpualongbj(s(j),r(j+1))islessthan,andthusthetotalchangeinargumentofpbualongbj(s(j),r(j+1))(whichofcourseequalsr(j+1))]TJ /F3 11.955 Tf 11.42 0 Td[(s(j))islessthan2inmagnitude(sincejpu)]TJ /F3 11.955 Tf 11.41 0 Td[(pbuj
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Weextendinthismannerto(s(j),r(j+1))foreachj2f1,...,N)]TJ /F4 11.955 Tf 12.1 0 Td[(1g.Moreover,wemayextendinusingtheexactlysimilarconstructionto[,r(1))and(s(N),],andthisextendedhasallofthedesiredproperties. CaseB.0.6. Thereisnosub-intervalof[,]alongwhichbtakesa((1),(1))trip.Theneitherjb(r))]TJ /F3 11.955 Tf 12.26 0 Td[(u(k)j(1)forallr2[,],orthereissomer(0)2(,)suchthatforallr2[,r(0)],jb(r))]TJ /F3 11.955 Tf 12.3 0 Td[(u(k)j(1),andforallr2(r(0),],bisgreaterthan(1)fromanycriticalpointofpu. SubcaseB.0.6.1. jb(r))]TJ /F3 11.955 Tf 11.96 0 Td[(u(k)j(1)forallr2[,].Inthiscase,weconstructusingthesamemethodasinthesecondpartofCase B.0.5 SubcaseB.0.6.2. Thereissomer(0)2(,)suchthatforallr2[,r(0)],jb(r))]TJ /F3 11.955 Tf 11.98 0 Td[(u(k)j(1),andforallr2(r(0),],bisgreaterthan(1)fromanycriticalpointofpu.Inthiscase,weconstructon[,r(0))usingthesamemethodasinthesecondpartofCase B.0.5 ,andweconstructon[r(0),]usingthesamemethodasintherstpartofCase B.0.5 Lemma10. Fixsomev=(v(1),...,v(n)]TJ /F5 7.97 Tf 6.58 0 Td[(1))2Vn)]TJ /F5 7.97 Tf 6.58 0 Td[(1notthezerovector,and(1)>0.Thenthereexistsconstants,(2)>0suchthatthefollowinghold.Letu2)]TJ /F5 7.97 Tf 6.59 0 Td[(1(v)bechosen,andxsomebu2B(u).Letbx1,bx22Gpbubegivensuchthatarg(pu(x1))=arg(pu(x2))=0,andsuchthatthereisapathb:[0,1]!Gpusuchthatd(0)=bx1andd(1)=bx2andarg(pbu(b(r)))=0forallr2[0,1].Thenifx1,x22Gpbuaresuchthatarg(pu(x1))=arg(pu(x1))=0andjbx1)]TJ /F3 11.955 Tf 12.67 0 Td[(x1j<(2)andjdx(2))]TJ /F3 11.955 Tf 12.67 0 Td[(x(2)j<(2),thenthereisapath:[0,1]!Gpusuchthat(0)=x1,(1)=x2,andforallr2[0,1],arg(pu((r)))=0andjb(r))]TJ /F6 11.955 Tf 12.78 0 Td[((r)j<(1).Moreover,ifjpbujisstrictlyincreasingorstrictlydecreasingonb,thenwemayassumethatjpujisstrictlyincreasingorstrictlydeacreasingonrespectively. 106

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Proof. TheexactsamemethodofproofusedforLemma 9 workshereexceptthatinsteadofinvokingLemma 6 wewouldinvokethegradientlineversionLemma 7 Lemma11. Fixsomev=(v(1),...,v(n)]TJ /F5 7.97 Tf 6.58 0 Td[(1))2Vn)]TJ /F5 7.97 Tf 6.59 0 Td[(1andsomeu2)]TJ /F5 7.97 Tf 6.59 0 Td[(1(v).Let>0begiven,andchoosesomepointx2Gpu.Thenthereareconstants,>0smallenoughsothatforanybu2B(u),ifby2B(pu(x))thenthereissomebx2B(x)suchthatpbu(bx)=by. Proof. NotethatthestatementofthelemmaissimilartothestatementofLemma 3 ,butmoregeneralinthatthepointxwhichischosenmaybeacriticalpointofpu.TheproofissimilartoaportionoftheproofofLemma 3 ,butwewillreproduceithere.ReduceifnecessarysothatB(x)Gpuandthereisnopointwsuchthatjw)]TJ /F3 11.955 Tf 11.33 0 Td[(xj=andpu(w)=pu(x).Thendene>0tobetheminimumthatjpu(w))]TJ /F3 11.955 Tf 11.33 0 Td[(pu(x)jtakesonthesetfw2C:jw)]TJ /F3 11.955 Tf 12.85 0 Td[(xj=g.Nowchoose>0sothatifbu2B(u),thenforallw2Gpu,jpu(w))]TJ /F3 11.955 Tf 12.62 0 Td[(pbu(w)j< 4.Deneh(z):=pu(x))]TJ /F3 11.955 Tf 11.95 0 Td[(pbu(z).Onthesetfw2C:jw)]TJ /F3 11.955 Tf 11.95 0 Td[(xj=gbyusingthereversetriangleinequalitywehavejh(z)j=jpu(x))]TJ /F3 11.955 Tf 10.56 0 Td[(pbu(z)j=jpu(x))]TJ /F3 11.955 Tf 10.57 0 Td[(pu(z)+pu(z))]TJ /F3 11.955 Tf 10.57 0 Td[(pbu(z)jjpu(x))]TJ /F3 11.955 Tf 10.56 0 Td[(pu(z)j+jpu(z))]TJ /F3 11.955 Tf 10.56 0 Td[(pbu(z)j,andthusjh(z)j)]TJ /F6 11.955 Tf 13.19 8.09 Td[( 2= 2.Howeverjh(x)j=jpu(x))]TJ /F3 11.955 Tf 12.74 0 Td[(pbu(z)j 4,sobytheMaximumModulusTheorem,weconcludethathcontainsazerointhesetfw:jw)]TJ /F3 11.955 Tf 12.53 0 Td[(xj0andR(0)2(0,1).Thereexistssome>0sothatthefollowingholds.Letu2K,andletz2Cbesuchthat 107

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jpu(z)j:=R2[R(0),1].Forallw2B(z),jpu(w)j2(R)]TJ /F6 11.955 Tf 12.01 0 Td[(,R+).Fixsomew2B(z),andletLdenotethestraightlinepathfromztow,thenjargj(pu,L)<. Proof. Reduceifnecessarysothat2(0,).SinceKiscompact,thereissomeS>0suchthatforeachu2K,GpuBS 2(0).AgainsinceKiscompact,andcl(BS 2(0))iscompact,thereissomeM>0suchthatforallu2K,forallz2BS(0),jpu0(z)j0sothatthefollowinghold.
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Lemma13. Let(f,G)beaspecialtypefunctionelement,andletbealevelcurveoffinG.LetDbesomefaceof,andletx,y2@Dbegivensuchthatthelinesegment(x,y)iscontainedentirelyinD.DeneR(0):=min(jf(z)j:z2[x,y]).DeneR(1):=max(jf0(z)j:z2[x,y]).Let+denotethepathinobtainedbytraversing@Dfromytoxwithapositiveorientation,andlet)]TJ /F9 11.955 Tf 10.41 -4.34 Td[(denotethepaththroughobtainedbytraversing@Dfromytoxwithanegativeorientation.Dene+:=arg(f,+),and)]TJ /F4 11.955 Tf 11.77 -4.18 Td[(:=arg(f,)]TJ /F4 11.955 Tf 7.09 -4.34 Td[().TheneithertherearezerosoffinDonbothsidesof[x,y],orjx)]TJ /F3 11.955 Tf 11.96 0 Td[(yj2R(0)min(j+j,j)]TJ /F2 11.955 Tf 7.09 -4.34 Td[(j) R(1). Proof. AssumethatallthezerosoffinDareononesideof(x,y)ortheother.Letdenotethepathobtainedbyconcatinating[x,y]with+.Thenisasimpleclosedpath.LetbDdenotetheboundedfaceof.AssumethatbDdoesnotcontainanyzerosoff.(Otherwise,wemaketheexactlysimilarargumentwith+replacedby)]TJ /F1 11.955 Tf 7.08 -4.34 Td[(.)SincebDdoesnotcontainanyzerooff,arg(f,)=0,andthusarg(f,[x,y])=)]TJ /F4 11.955 Tf 9.3 0 Td[(arg(f,+),andthusjarg(f,[x,y])j=j+j.Letdenotethestandardparameterizationof[x,y].Let0=t(0)<
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jz(i))]TJ /F3 11.955 Tf 11.96 0 Td[(z(i)]TJ /F5 7.97 Tf 6.59 0 Td[(1)j2R(0)j+ij R(1).Andifwesumoveralli,weobtainjx)]TJ /F3 11.955 Tf 11.96 0 Td[(yj2R(0) R(1)NXi=1j+ij2R(0)j+j R(1)2R(0)min(,j)]TJ /F2 11.955 Tf 7.08 -4.33 Td[(j) R(1),whichisthedesiredresult. Lemma14. Let(f,G)beaspecialtypefunctionelement,andletbealevelcurveoffinG.Letx,y2begivensuchthatthelinesegment(x,y)iscontainedentirelyintheunboundedfaceof.LetbDdenotetheboundedfaceof[[x,y]whichisnotaboundedfaceof.Let(0)beaparameterizationoftheportionof@bDwhichisinfromytox,anddene:=arg(f,(0)).DeneR(0):=min(jf(z)j:z2[x,y]).DeneR(1):=max(jf0(z)j:z2[x,y]).TheneitherbDcontainszerosoff,orjx)]TJ /F3 11.955 Tf 11.96 0 Td[(yj2R(0)jj R(1). Proof. BythesamereasoningfoundinLemma 13 110

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REFERENCES [1] W.K.Hayman,J.M.G.Wu,Levelsetsofunivalentfunctions,Comment.Math.Helv.3(1981)366. [2] P.W.Jones,Boundedholomorphicfunctionswithalllevelsetsofinnitelength,Mich.Math.J.(1980)75. [3] P.Erdos,F.Herzog,G.Piranian,Metricpropertiesofpolynomials,J.AnalyseMath.6(1958)125. [4] A.W.Goodman,Ontheconvexityofthelevelcurvesofapolynomial,Proc.Amer.Math.Soc.17(1966)358. [5] G.Piranian,Theshapeoflevelcurves.,Proc.Amer.Math.Soc.17(1966)1276. [6] M.G.Valiron,Surlescourbesdemoduleconstantdesfonctionsentieres,C.R.Acad.Sci.Paris204(1937)402. [7] K.Stephenson,C.Sundberg,Levelcurvesofinnerfunctions,Proc.LondonMath.Soc.(3)51(1985)77. [8] K.Stephenson,Analyticfunctionssharinglevelcurvesandtracts,Ann.ofMath.(2)123(1986)107. [9] A.A.Kirillov,Kahlerstructureonthek-orbitsofagroupofdiffeomorphismsofthecircle,FunktsionalAnal.iPrilozhen21(1987)42. [10] A.A.Kirillov,Geometricapproachtodiscreteseriesunirrepsforvir,J.Math.PuresAppl.(9)77(1998)735. [11] P.Ebenfelt,D.Khavinson,H.S.Shapiro,Two-dimensionalshapesandlemniscates,Contemp.Math.553(2011)45. [12] J.Conway,FunctionsofonecomplexvariableII,Springer,NewYork,1995. [13] J.Conway,FunctionsofonecomplexvariableI,2ed.,Springer,NewYork,1978. [14] A.F.Beardon,T.K.Carne,T.W.Ng,Thecriticalvaluesofapolynomial,ConstructiveApproximations18(2002)343. [15] S.Roman,Theumbralcalculus,Dover,NewYork,2005. 111

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BIOGRAPHICALSKETCH TrevorRichardswasborninLansing,Michiganin1983.TrevormovedtoFloridain2002topursueunder-graduatestudiesattheUniversityofFlorida,andgraduatedwithaBSinmathematicswithaminorinphilosophyin2006.TrevorbegangraduatestudyintheUniversityofFloridaMathematicsDepartmentin2007. 112