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The Free Grothendieck Theorem

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Title:
The Free Grothendieck Theorem
Creator:
Augat, Meric L
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[Gainesville, Fla.]
Florida
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University of Florida
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english
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1 online resource (77 p.)

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Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Mathematics
Committee Chair:
Mccullough,Scott A
Committee Co-Chair:
Pascoe,James Eldred
Committee Members:
Robinson,Paul L
Keesling,James E
Jury,Michael Thomas
Webster,Gregory Daniel
Graduation Date:
5/3/2019

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free-analysis -- free-polynomials -- inverse-function-theorem -- jacobian-matrix -- nc-jacobian
Mathematics -- Dissertations, Academic -- UF
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theses ( marcgt )
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Mathematics thesis, Ph.D.

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Abstract:
The main result of this article establishes the free analog of Grothendieck's Theorem on bijective polynomial mappings of C^n. Namely, we show if p is a free polynomial mapping in g freely noncommuting variables sending g-tuples of matrices (of the same size) to g-tuples of matrices (of the same size) that is injective, then it has a free polynomial inverse. Other results include an algorithm that tests if a free polynomial mapping p has a polynomial inverse (equivalently is injective; equivalently is bijective). Further, a class of free algebraic functions, called hyporational, lying strictly between the free rational functions and the free algebraic functions are identified. They play a significant role in the proof of the main result. ( en )
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Thesis (Ph.D.)--University of Florida, 2019.
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Adviser: Mccullough,Scott A.
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Co-adviser: Pascoe,James Eldred.
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by Meric L Augat.

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THEFREEGROTHENDIECKTHEOREM By MERICLANGSTONAUGAT ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF DOCTOROFPHILOSOPHY UNIVERSITYOFFLORIDA 2019

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c 2019MericLangstonAugat

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Toallthosewholeadmonotonouslives,inthehopethattheymayexperienceatsecondhand thedelightsanddangersofmathematics.

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ACKNOWLEDGMENTSTheauthorwouldliketothankhisadvisorScottMcCulloughforthemultiplereviewsofthismanuscriptandforthemanyvaluablediscussionsalongtheway.TheauthorwouldalsoliketothankIgorKlep,JurijVolcic,JamesPascoeandSpelaSpenkoformanyhelpfulconversationsandforpointingouttheexistenceofseveralalgebraicresults. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS...................................4 ABSTRACT.........................................6 CHAPTER 1INTRODUCTION...................................7 TheJacobian,FreeAlgebraicFunctionsandProperAlgebraicSystems.......9 HyporationalFunctions................................10 Reader'sGuide.....................................12 2PRELIMINARIES...................................15 3JACOBIANMATRICESANDFREEANALYTICFUNCTIONS............18 TheJacobianMatrixofaFormalPowerSeries....................18 AuxiliaryInversesandCompositionalInverses.....................20 InvertibilityoftheJacobianMatrix..........................28 4FREEDERIVATIVESANDLINEARIZATIONS....................32 PolynomialCriteria..................................32 FreeDerivativesandScions..............................34 5DEGREEBOUNDSONNCRATIONALMAPS...................41 RationalDegreeBounds................................41 GenericMatrixAlgebras................................47 6HYPORATIONALSERIES...............................50 HyporealizationsandHyporationalSeries.......................50 HypomatrixRepresentations..............................52 BijectivityCriteria...................................60 7COMPUTINGINVERSES...............................65 APPENDIX:ACOLLECTIONOFPROOFS........................70 REFERENCELIST.....................................74 BIOGRAPHICALSKETCH.................................77 5

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AbstractofDissertationPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulllmentofthe RequirementsfortheDegreeofDoctorofPhilosophy THEFREEGROTHENDIECKTHEOREM By MericLangstonAugat May2019 Chair:ScottA.McCullough Cochair:JamesE.Pascoe Major:MathematicsThemainresultofthisarticleestablishesthefreeanalogofGrothendieck'sTheoremonbijectivepolynomialmappingsofC g.Namely,weshowifpisafreepolynomialmappingingfreelynoncommutingvariablessendingg-tuplesofmatricesofthesamesizetog-tuplesofmatricesofthesamesizethatisinjective,thenithasafreepolynomialinverse.Otherresultsincludeanalgorithmthattestsifafreepolynomialmappingphasapolynomialinverseequivalentlyisinjective;equivalentlyisbijective.Further,aclassoffreealgebraicfunctions,calledhyporational,lyingstrictlybetweenthefreerationalfunctionsandthefreealgebraicfunctionsareidentied.Theyplayasignicantroleintheproofofthemainresult. 6

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CHAPTER1 INTRODUCTIONAremarkablepairoftheoremsofGrothendieck[10,11]sayifp:C g ! C gisaninjectivepolynomial,thenpisbijectiveanditsinverseisapolynomial.Degreeboundsontheinversewerediscoveredsoonafter.TheseresultsareofcourseintimatelyconnectedwiththefascinatingJacobianconjectureseeforinstance[2,9]andthequestionoftameversuswildautomorphismsofthepolynomialringseeforinstance[28,29,32,33].InthisarticleweprovethefreeGrothendiecktheorem.OurapproachinvolvescarefulanalysisofthenoncommutativeJacobianmatrixasfoundin[25],thetheoriesoffreerationalseries[3]andtheirrealizations[20,34],formalpowerseriesinnoncommutingvariables[30],freeanalysis[12,14],properalgebraicsystems[24,26],freederivatives[23]andskewelds[4].Wealsomakeuseofsomenewmachineryincludingthehyporationalfunctionsandthehypo-Jacobianmatrixdenedlaterinthispaper.Tostatetheresult,xapositiveintegergandletx=x 1 ;:::;x gdenoteatupleoffreelynoncommutingindeterminants.Afreepolynomialing-variablesisaniteClinearcombinationofwordsinx.Forpositiveintegersn,letM nCgdenotetheg-tuplesofn nmatricesoverCandletMCgdenotethesequenceM nCgn.Afreepolynomialfinducesasequenceofmapsf[n]:M nCg ! M nCbyevaluation,X 7! fX.Weletf:MCg ! MCdenotethissequence.Afreepolynomialmapping p:MCg ! MCgisthusag-tupleofsequences,p=p 1 ;:::;p g,thatis,eachp iisafreepolynomial.Thepolynomialmappingpisinjectiveresp.surjective,bijectiveifeachp[n]isinjectiveresp.surjective,bijective.Ofcourseifp[n]isinjective,thenconsideredasapolynomialingn 2commutingvariables,itisbijectiveandhasapolynomialinverse.ThefollowingfreepolynomialanalogofGrothendieck'sTheoremwasimplicitlyconjecturedin[23]. Theorem6.11FreeGrothendieckTheorem.Ifp:MCg ! MCgisaninjectivefreepolynomialmapping,thenthereisafreepolynomialmappingqsuchthatp qx=x=q px;thatis, p hasafreepolynomialinverse. 7

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Beforedescribingourmethodsinfurtherdetail,wepausetonotethatTheorem6.11isofcourserelatedtothestudyofautomorphisms,andthequestionoftameversuswildautomorphismsofthefreealgebraseeforinstance[5,7,8,21,31].Pascoe[23]provesafreefreelynoncommutativeinversefunctiontheoremandusesthistheoremtoestablishafreeanalogoftheJacobianconjecture,statedbelow. TheoremAFreeJacobianConjecture.Supposep:MCg ! MCgisafreepolynomialmapping.Thefollowingareequivalent: i DpX[H]isnonsingularforeachX 2 MCg;thatis,foreachpositiveintegernandeachtuple X 2 M n C g ,thelinearmapping M n C g 3 H 7! Dp X [ H ] isnon-singular; ii p isinjective; iii p isbijective; iv p )]TJ/F28 7.9701 Tf 6.587 0 Td [(1existsasafreefunction,andforeachn,p )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 j M n C gagreeswithafreepolynomialmapping;namely,thereexistsafreemappingg:MCg ! MCgandfreepolynomialmappingsq nsuchthatpgX=X=gpXforallX 2 MCgandgX=q nXforeach n and X 2 M n C g :ThenotionofafreefunctionisdenedinChapter2andthefreederivativeDpisdenedinSection4.Wewilloftenusetheequivalenceofitemsiiandiii.Threeresultsinthisarticlerequirelittleornoadditionaloverheadtostate.Assumingpisinjective,Theorem7.2producesboundsforthedegreeofitsinverseq.Asaconcreteexample,degqg Q g i =1 i 3degp)]TJ/F25 11.9552 Tf 1.02 0 0 1 231.961 253.796 Tm [(1+1.Usingthedegreebound,Corollary7.1describesanalgorithmthattakesasinputafreepolynomialpand,afteranumberofiterationsdependingonlyonthenumberofvariablesgandthedegreeofp,eitherproducesapolynomialq-theinverseof p -or p isnotinjectiveandhasnopolynomialinverse.Thederivative,Dpx[h],ofafreepolynomialpisag-tupleofpolynomialsinthe2gfreelynoncommutingvariablesh ; x=h 1 ;:::;h g ;x 1 ;:::;x gdenedasthefreeanalogofthedirectionalderivativeintheobviousway.TheresultofPascoementionedabove{p 8

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isbijectiveifandonlyifh 7! Dpx[h]ispointwisenon-singular{isstrengthenedbythefollowingresult. Corollary6.1.Afreepolynomialp:MCg ! MCgisbijectiveifandonlyifh ; x7! Dp x [ h ] ; x hasapolynomialinverse.WeusethefreederivativetostateandproveTheorem4.5,theimplicitfunctiontheoremforncformalpowerseries.ItismostlyaconsequenceofLemma3.3.Wereferto[1]foranindepthanalysisoftheimplicitfunctiontheoremforseveraltopologieson M C g . Theorem4.5Implicitfunctiontheorem.Supposefx ; z2C t x [ z yh.Iff;0=0and@f=@ z;02 M hCisinvertible,thenthereexistsauniqueg 2C t x yhsuchthatg =0 and f x ; g x =0 . TheJacobian,FreeAlgebraicFunctionsandProperAlgebraicSystemsTheleftJacobianmatrix[25]ofafreepolynomialmappingp:MCg ! MCgwithnoconstanttermistheunique g g matrix J p withfreepolynomialentriessuchthat p x = p 1 x p g x = x J p x = x 1 x g J p x : Inparticular, p j x = g X s =1 x s J p s;j x :ThedenitionoftheJacobianmatrixJ pextendsnaturallytothecasewhereeachp jisafreeformalpowerserieswithnoconstantterm.InthiscaseJ pisag gmatrixwithfreeformalpowerseriesentries.Ithasamultiplicativeinverseifthereisag gmatrixJ poffreeformalpowerseriessuchthatJ pxJ px=I gandJ pxJ px=I g.InthiscaseJ pisuniqueanddenoted J )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 p .ThefollowingpropositioncombinesCorollary3.1andLemma3.3. Proposition3.2.Supposepisafreeformalpowerseriesmappingwithoutconstantterm.ThereisafreeformalpowerseriesmappingqwithoutconstanttermsuchthatpandqarecompositionalinversesifandonlyifJ phasafreeformalpowerseriesmultiplicativeinverse.9

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Inthiscase, J p q x = J )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 q x .Moreover, q istheuniquesolutionof q x = x J )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 p q x : -6InthecasethatJ )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 pisapolynomial,Equation3-6impliesq,theinverseofp,isalgebraic.Tostatetheresultmorepreciselyrequiresadenition.Supposez=z 1 ;:::;z hisanadditionaltupleoffreelynoncommutingvariablesandx[z]= 1 h isapolynomialmapping.Wesayisaproperalgebraicpolynomialmappingifhasnoconstanttermsandeachmonomialappearinginwithdegreeinzofatleastone,hastotaldegreeofatleasttwo.Atupleoffreeformalpowerserieswithoutconstantterm, x = 1 h ,isa solution totheproperalgebraicpolynomial if x [ x ]= x :Wesayeach iisacomponentofthesolution.By[30,Theorem6.6.3],everyproperalgebraicpolynomialmappinghasauniquesolution.Aformalpowerseriesxisalgebraicif )]TJ/F30 11.9552 Tf 11.955 0 Td [(c isacomponentofthesolutiontosomeproperalgebraicpolynomialmapping.Ifbothpandp )]TJ/F28 7.9701 Tf 6.586 0 Td [(1areapolynomialmappings,thenthechainruleimpliesJ )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 pisapolynomialmatrixRemark3.4and[25,Corollary1.4].ThusthefollowingtheoremfollowsimmediatelyfromTheorem6.11.WegiveanindependentproofandtheresultitselfisakeyingredientintheproofofTheorem6.11. Theorem3.5. If p isabijectivepolynomialmapping,then J )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 p isapolynomialmatrix.Example3.7conciselypointsoutthelimitationsoftheJacobianmatrix;itdoesnotdetectthenon-injectivityofapolynomial. HyporationalFunctionsIfpisabijectivefreepolynomial,thennecessarilyitsinverseqisanalgebraicmapping.Ifinaddition,qisrational,then[20,Theorem4.2]impliesqisapolynomial.InChapter6weidentify,intermsofproperalgebraicpolynomialmappings,freerationalfunctionsamongstfree10

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algebraicmappingsandextend,inTheorem6.7below,[20,Theorem4.2]toalargerclassoffreealgebraicfunctions.Infact,aformalpowerseriesisrationalifandonlyifitisacomponentofthesolutionofsomeproperalgebraicpolynomial oftheform, x [ z ]= a x + z A x ;whereaisapolynomialmappingandAisapolynomialmatrix.Ontheotherhand,example6.3showsthatthesolutiontoaproperalgebraicpolynomialmappinghavingdegreeoneinzisnotnecessarilyrational.Aformalpowerseriesaxwithconstantterma 1isahyporationalseriesifa )]TJ/F30 11.9552 Tf 12.004 0 Td [(a 1isacomponentofthesolutiontoaproperalgebraicpolynomialmappingx[z]ofdegreeoneinz.EveryrationalseriesisahyporationalseriesandExample6.3showsthisinclusionisproper.Hencehyporationalfunctionslieproperlybetweenfreerationalfunctionsandfreealgebraicfunctions.Thefollowingresultshowsthathyporationalsenjoysomeofthesameregularitypropertiesasrationals. Theorem6.7.Supposeaishyporational.Ifdom na=M nCgforalln,thenaisafreepolynomial.InChapter6weintroducethehypo-Jacobianmatrix J hyp pofafreepolynomialmappingp.Itisag gmatrixwhoseentriesarebipartitepolynomials;thatispolynomialsinthetwog-tuplesoffreelynoncommutingvariablesxandy,butwherex j y k=y k x jforall1 j;k g.SeeLemma6.3andDenition6.4.Theorem6.9showsthatthehypo-Jacobianmatrixofafreepolynomialmappingissimplyamatrixformofthefreederivative;thehypo-Jacobian'sinvertibilityasamatrixencodestheinvertibilityoffreepolynomials.Indeed,weobtainthefollowingimprovementofTheorem3.5. Theorem6.9.ThefreepolynomialmappingpisinjectiveifandonlyifJ hyp phasamultiplicativeinversewhoseentriesarebipartitepolynomials. 11

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Inotherwords,afreepolynomialmappingpisinjectiveifandonlyifitshypo-Jacobianmatrixisinvertibleasabipartitepolynomialmatrix.Thenotionofthehypo-JacobianmatrixarisesinthestudyofendomorphismsofthefreeassociativealgebraC h x i.Infact,anysuchendomorphismhasaJacobianmatrixsee[6]and[27]thatexactlycorrespondswithournotionofthehypo-Jacobianmatrix. Reader'sGuideChapter2introducesdenitionsandnotationfromformalpowerseriesandfreeanalysisthatarerepeatedlyusedthroughoutthepaper.TheJacobianmatrixofaformalpowerseriesisdenedinChapter3anditservesasoneofthecentralobjectsofstudy.InvertibilityoftheJacobianmatrixisnecessaryandsucientforaformalpowerseriesmappingtohaveacompositionalinverseProposition3.2.InSection3weborrowideasfromenumerativecombinatorics-namelytheconstructionofanalgebraicformalpowerseriesbyiteratingthecompositionofasetofpolynomials-andexploitthechainrulefortheJacobianmatrixtoiterativelyconstructthesecompositionalinverses.Section3extendsresultsaboutJacobianmatricestofreeanalyticfunctions.TheseresultsarethencombinedwithanoncommutativeNullstellensatz{duetoman[14]{toprovethekeyintermediateresultTheorem3.5:ifafreepolynomialisinjective,thenitsJacobianmatrixhasapolynomialmatrixinverse.Section4exploressimpleconditionsthatguaranteeafreepolynomialhasafreepolyno-mialinverse.WhileinSection4werecallthefreederivativeasdenedin[23]andinvestigateitsproperties.Foraxedfreepolynomialp,wedenethefunctionF:x ; y7!Dpy[x]; yandobservethatPascoe'ssolution[23]tothefreeJacobianconjecturecanbeinterpretedassaying,pisbijectiveifandonlyifFisbijective.SettingGequaltothefreeinverseofF,Lemma4.2showsthatthereisaz-anelinearmappingGsuchthattherstgentriesofG arethesolutionto G .Understandingtheinversefunction G iswhatmotivatesChapter6.Chapter5seeminglydepartsfromthepreviousdiscussionandestablishesfactsaboutnoncommutativerationalfunctionsandrationaldegreemapsneededinthefollowingchapter.12

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ThemainresultofthischapterisProposition5.1.Itshowsthatevaluatingancrationalfunctionronmatricesproducesamatrixwhoseentriesbehavemuchliketheabelianizationofr .Chapter6introducesthehyporationals,ageneralizationofrationalformalpowerseries.Weproceedtoshowthatthefreederivativeofaninjectivepolynomialhasahyporationalinverse.Ifsisahyporationalseriesthatisnotrational,thenwecannotapplyresultsfromrealizationtheory.However,s[n]=s j M n C gisacommutativerationalfunctionforeachn,hintingthatitmaybepossibletoextendregularityresultsfromncrationalfunctionstohyporationalfunctions.Proposition6.1doessobyconstructingthehypomatrix representationofhyporationalfunction;amatrixoverC h x i C h y i,thealgebraofbipartitepolynomials,thatimitatestherealizationtheoryofncrationalfunctions.ThealgebraC h x i C h y iiscontainedinaskeweldC < x y >,andthehypomatrixrepresentationisinvertibleasamatrixoverC < x y >.Thus,wemayusetheresultsofChapter5toanalyzehyporationalfunctions.ByapplyingProposition5.1weproveTheorem6.7:ahyporationalfunctionwithnodomainexceptionsisinfactapolynomial,aresultestablishedin[20]and[17]forfreerationalfunctions.AstraightforwardconsequenceisCorollary6.1.ItstrengthensPascoe'sresolutionoftheFreeJacobianConjecturebyasserting:afreepolynomialpisinjectiveifandonlyifh ; x7!Dpx[h]; xhasapolynomialinverse.Thiscorollaryisbothaningredientin,andimmediateconsequenceof,Theorem6.11assumingbijectivityof F x ; y = Dp y [ x ] ; y .InSection6weintroduceJ hyp p,thehypo-Jacobianmatrixofthefreepolynomialp.UsingCorollary6.1weproveTheorem6.9:afreepolynomialisinjectiveifandonlyifitshypo-Jacobianmatrixisinvertibleasamatrixofbipartitepolynomials.ConnectingTheorem6.9toresultsin[6]and[27]provestheFreeGrothendieckTheorem,Theorem6.11.Lastly,inChapter7wediscusscomputationalaspectsofcomputingthefreeinverseqofagivenfreepolynomialp.IfqisafreepolynomialthenTheorem7.2providesanupperbound13

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forthedegreeofqdependingonlyonthenumberofvariablesandthedegreeofp,leadingtoanalgorithmictestforwhether p hasapolynomialinverse,Lemma7.3. 14

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CHAPTER2 PRELIMINARIESLetAbeanyC-algebra.Wedenotethen nmatrixalgebrawithentriesinAbyM nA.Letx=f x 1 ;:::;x g gbeasetofnoncommutingindeterminates.Thesetofnitesequencesofelementsofxisdenotedbyh x i.Theemptysequenceistheidentityelementofh x iandisdenotedby 1 .Anelementofxiscalledaletter,anelementofh x iiscalledawordandthelengthofawordw=x i 1 :::x i mism,denotedbyj w j.Wedenotethealgebraoffreeformalpower series withcoecientsin A by A t x y andif f 2A t x y then f = X w 2h x i c w w; whereeach c w 2A .Wesayp 2A t x yisapolynomialifallbutanitenumberofthecoecientsofparezero.Thesetofallpolynomials,denotedAh x i,isthefamiliarfreealgebraongnoncommutingindeterminates.ItisasubalgebraofA t x y.Wedenotetheformalpowerserieswithnoconstanttermby A t x y + andtheformalpolynomialswithnoconstanttermby Ah x i + . Suppose = P w a w w and = P w b w w .Dene ! : A t x y A t x y ! N [f1g by ! ; =inf f n 2 N : 9 w 2h x i ; j w j = n and a w 6 = b w g :Thefunctiond:A t x y A t x y ! Rgivenbyd;=2)]TJ/F31 7.9701 Tf 6.586 0 Td [(! ; isametriconA t x y.Furthermore,A t x yiscompleteandAh x iisdenseinA t x y.Themetrictopologyaboveisequivalenttothe x -adictopology.FormalpowerseriesmaybegeneralizedfurthertofreeproductsofunitalC-algebras.Aneasyexampleofsuchapowerseriesisapolynomialp 2 C h x [ z i,whichcaninsteadbetakenasapolynomialinthefreeproductofC h x iwithC h z i.ThefreeproductofC h x iandC h z iisthesetofallwords 1 1 ::: k k,where i 2h x i, i 2h z iarenonemptywords.Amuchmoredetailedexpositioncanbefoundin[34]. 15

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Denition2.1.IfS 2 C t x yandShasanonzeroconstantterm,thenS )]TJ/F28 7.9701 Tf 6.587 0 Td [(1themultiplicativeinverseof S ,existsandisgivenby S )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 = 1 X n 0 1 )]TJ/F30 11.9552 Tf 13.151 8.088 Td [(S ! n :LetC rat t x ydenotethealgebraofrationalseries;thesmallestsubalgebraofC t x y containing C h x i suchthatif S 2 C rat t x y and S )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 exists,then S )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 2 C rat t x y .Wegivebasicdenitionsandafewresultsinfreeanalysisthatwillbeusedthroughoutthepaper.Afreepolynomialisanoncommutativepolynomialevaluatedontuplesofmatricesthatpreservesthestructureoffreesets.Afreeset)-303(=[n]1 n =1 MCg=M nCg1 n =1isagradedsetoftuplesofmatricesthatisclosedunderdirectsumsandconjugationbysimilarities.Thatis,if X 2 [ n ] , Y 2 [ m ] and S 2 GL n C then i X Y 2 [ n + m ] ; ii S )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 XS 2 [ n ] , where X Y = X 1 ;:::;X g Y 1 ;:::;Y g = X 1 Y 1 ;:::;X g Y g and S )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 XS = S )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 X 1 S;:::;S )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 X g S :LetU MCgbeafreeset.AfreemaporfreefunctionffromUintoMCisasequenceoffunctionsf[n]:U[n]! M nCthatrespectsthefreestructureofU;fX Y=fX fYandfS )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 XS=S )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 fXS.Thenotionofafreemapextendseasilytovector-valuedfunctionsf:U ! MCh,matrix-valuedfunctionsf:U ! M dMCgandevenoperator-valuedfunctions.Iff i:U ! MCisafreemapfor1 i hthenwesaythetuplef=f 1 ;:::;f hisafreemappingandwritef : U ! M C h . 16

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SupposeU MCgisafreesetandeachU[n]isopenasasubsetofM nCg.Afreemapf:U ! MCiscontinuousifeachf[n]iscontinuous,andisfreeanalyticifeachf[n]isanalytic.Asshownin[16],afreefunctionthatiscontinuousisalsofreeanalyticseealso[12].Asonewouldhope,thereareindeedconnectionsbetweenfreeanalyticfunctionsandformalpowerseries.Infact,aformalpowerserieswithapositiveradiusofconvergencedeterminesafreeanalyticfunctionandwithasmalldegreeoflocalboundednesswegettheconversesee[13]. Givenapositiveinteger d let f 2 M d C h t x y ,thatis f = 1 X m =0 X w 2h x i j w j = m c w w;wherec w 2 M dCh.ForX 2 M nCgandw=x i 1 :::x i k 2h x i,wesaywX=X w=X i 1 :::X i k .Wedenetheevaluationof f at X by f X = 1 X m =0 X w 2h x i j w j = m c w X w ; providedthisseriesconverges. 17

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CHAPTER3 JACOBIANMATRICESANDFREEANALYTICFUNCTIONSFixg 2 Z +andsetx=x 1 ;:::;x g2h x i g,withxconsideredasarowvector.Forh 2 Z +and1 i h,let i 2 C t x yand= 1 ;:::; h2C t x yh.Alternatively,wecanview asanelementof C h t x y ,thesetofformalpowerserieswithcoecientsin C h .Leth;k 2 Z +.Supposey=f y 1 ;:::;y h gandz=f z 1 ;:::;z k garesetsoffreelynoncommutingindeterminatesandsupposehasnoconstantterm,thatis, 2C t x y +h.Inthiscase,wemayviewfromamuchmorealgebraicperspective;:C t y y ! C t x yisacontinuoushomomorphismdenedby y i 7! i . TheJacobianMatrixofaFormalPowerSeriesWedenetheleftnoncommutativeJacobianmatrixofaformalpowerseries,acentralobjectofstudythroughoutthepaper.AtreatmentofthenoncommutativeJacobianmatrixcanbefoundin[25]. Denition3.1.Letf 2 C t x ywithf=P w 2h x i f w w.For1 i g,deneS x i:C t x y ! C t x y by S x i f = X w 2h x i f x i w w: Inotherwords, S x i istheadjointoftheoperatorofleftmultiplicationby x i .Leth 2 Z +andtake 2C t x yh,seenasarowvectorofformalpowerseries.Theg h matrixover C t x y denedby J = S x j i g;h i;j =1istheleftJacobianmatrixof.Inparticular,ifhasconstantterm 1= 1 1 1 ;:::; h 1 1,then = 1 + x J ;wherex J isthestandardproductofarowvectorandamatrix.Thisrepresentationofisunique.Remark3.1.Let 2C t x y +handdenethehomomorphism:C t y y ! C t x yby y i = i .Dening J = J yieldstheJacobianmatrixencounteredin[25]. 18

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ItisevidentthateveryformalpowerserieshasaJacobianmatrixandif; 2C t x yh havethesameJacobianmatrixthen )]TJ/F30 11.9552 Tf 11.955 0 Td [( 2 C h .Remark3.2.If:C t y y ! C t x yand:C t z y ! C t y yarecontinuoushomomorphisms,thencertainly :C t z y ! C t x yisacontinuoushomomorphism.Astuplesofformalpowerseriesthissaysthatx2C t x y +k.Thisalignswiththefactthatxisdenedaslongas hasazeroconstantterm.ForanyA 2 M k nC t y yand 2C t x yh,A=AyandAx2 M k nC t x y,whereAxistheresultofapplyingthehomomorphismy i 7! itoeachentryofA.Thus,ifB 2 M n mC t y ythenAxBx=ABx2 M k mC t x y.Inparticular,ifC 2 M n C t y y isaninvertiblematrixthen C )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 = C y )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 and C )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 x = C x )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 . Proposition3.1. If 2 C t x y + h and 2 C t y y + k then J x = J x J x 2 M g k C t x y : Proof.Observe 2C t x yk.Dene:C t y y ! C t x yand:C t z y ! C t y ybyy i= iandz j= j.Thus, :C t z y ! C t x ywith z i= i.ByProposition1.2in[25], J = J x J x 2 M g k C t x y . Corollary3.1.Supposep;q 2C t x y +ghaveJacobianmatricesJ pandJ q,respectively.TheseriespandqarecompositionalinversesifandonlyifJ pqx=J qx)]TJ/F28 7.9701 Tf 6.587 0 Td [(1andJ q p x = J p x )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 . Proof.Supposepandqarecompositionalinverses.Hencepqx=xandqpx=x.ApplyingProposition3.1, J q x J p q x = J p q x = I g and J p x J q p x = J q p x = I g : Thus J p q x = J q x )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 and J q p x = J p x )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 . 19

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Nowsuppose J p q x = J q x )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 and J q p x = J p x )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 .Observe p q x = x J p q x = x J q x J p q x = x I g = x ; and q p x = x J q p x = x J p x J q p x = x I g = x : Therefore, p and q arecompositionalinverses. TheinvertibilityoftheJacobianmatrixisreminiscentoftheinversefunctiontheorem.Indeed,ifp 2C t x y +gisafreefunction,thenpislocallyinvertibleat0ifandonlyifJ pisinvertibleat 0 .ItshouldbenotedthatJ )]TJ/F29 5.9776 Tf 5.756 0 Td [(1x=J )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 )]TJ/F28 7.9701 Tf 6.586 0 Td [(1x,hencewecannotuseJ todirectlycompute )]TJ/F28 7.9701 Tf 6.587 0 Td [(1withoutalreadyknowingtheexplicitformof )]TJ/F28 7.9701 Tf 6.586 0 Td [(1.However,J )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 isalocalat0approximationofJ )]TJ/F29 5.9776 Tf 5.756 0 Td [(1,implyingwemaybeabletoconstruct )]TJ/F28 7.9701 Tf 6.586 0 Td [(1fromsuccessiveapproximations.ThisleadsusdirectlytoSection3. AuxiliaryInversesandCompositionalInversesThemainresultinthissection,Proposition3.2,tellsusthatifp 2C t x yg,suchthatJ p isinvertible,then p )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 isthelimitofasequenceofpolynomialsconstructedfrom J )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 p . Denition3.2.Supposeh 2 Z +andz=f z 1 ;:::;z h gisasetoffreelynoncommutingindeterminates.Foranyw 2h x [ z idenej w j ztobethenumberofz-termsappearinginw anddene j w j x tobethenumberof x -termsappearingin w .Inparticular, j w j = j w j x + j w j z . Let ` 2 Z + and 2 C ` t x [ z y + = C t x [ z y + ` with = P w 2h x [ z i a w w .Dene d z =inf fj w j : j w j z > 0 and a w 6 =0 g : -1 Noteif hasno z -termsthen d z = 1 .Wewillconsistentlywriteaformalpowerseries 2C t x [ z y`asx[z]ratherthanx ; z.Thisconventionissimplyapreferencebasedonaligningournotationwiththenotationweuseforfreederivatives.Thus,if 2C t x [ z y +hthenx[x[z]]writtenanotherwayis x ; x ; z . 20

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Lemma3.1.Let; 2 C ` t x [ z y +and 2 C h t x [ z ywith=P w 2h x [ z i a w w, = P w 2h x [ z i b w w ,and = P w 2h x [ z i c w w .Wehavethefollowing, i d z + min f d z ; d z g ; ii d z min f d z ; d z gandinparticular,ifd zandd zarenotbothinnite,then d z > min f d z ; d z g ; iiiif d z > 1 and d z < 1 then d z < d z x [ x [ z ]] . Proof.Sincea w+b wisnonzeroonlyifatleastoneofa worb wisnonzero,weautomaticallyhave d z + min f d z ; d z g .Thuswehaveproven i .Toproveii,rstsupposed z; d z=1,i.e.neitherhasaz-term.Itfollowsthattheirproduct, hasno z -termsandthus d z = 1 .Nowsupposed zisnite.Inthecasethatd zisinnite,weseed z> d z min f d z ; d z gandwearedone.Finally,supposed z< 1andlet=P w 2h x [ z i c w w.Letwbeamonomialwithj w j z >0,j w j=d zandc w 6=0.Thereexistmonomialsu;vsuchthata u b v 6=0anduv=w.Recallandhavenoconstanttermsso j u j ; j v j > 0 .Since j w j z > 0 wemayassume j u j z > 0 ,hence j u j d z and d z = j w j = j u j + j v j d z + j v j > d z min f d z ; d z g : Thus, d z > min f d z ; d z g anditemiiisdone. Toproveitemiii,suppose d z > 1 and d z < 1 .Set W = f w 2h x [ z i : j w j z > 0 ;a w 6 =0 gandnoteifWisemptythend zx[x[z]]=1 > d z.SupposeWisnonempty.Writing x [ x ]= X w 2h x [ z i a w w x [ x [ z ]] andapplyingitemiyields d z x [ x [ z ]] min f d z a w w x [ x [ z ]]: w 2 W g : 21

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Supposethereisaw 2 Wsuchthatd za w wx[x[z]]< 1otherwisewehavethatd zx[x[z]]=1andwearedone.Weknowj w j2sincew 2 Wandd z2.Hencewx[x[z]]isaniteproductofx iand ix[z]terms.Thus,forw 2 W , d z a w w x [ x [ z ]] > d z ,andtherefore d z x [ x [ z ]] > d z . Lemma3.2. Suppose a 2 C t x [ z y + h anddene a k x [ z ]= a x [ a k )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 x [ z ]] for k> 1 and a 1 x [ z ]= a x [ z ] .If n;m 2 Z + then a n x [ a m x [ z ]]= a m x [ a n x [ z ]]= a n + m x [ z ] : Proof. Werstprove a n +1 x [ z ]= a x [ a n x [ z ]]= a n x [ a x [ z ]] -2viainduction.Thebasecaseisfromthedenition,sosupposeEquation3-2holdsfornandconsider a n +2 x [ z ]= a x [ a n +1 x [ z ]]= a x [ a n x [ a x [ z ]]] = a n +1 x [ a x [ z ]] : ThusEquation3-2holdsingeneral.Now,takeanyn;m 2 Z +andconsidera nx[a mx[z]].ApplyingEquation3-2mtimesyieldsa nx[a mx[z]]=a n + mx[z];whileapplyingEquation3-2ntimesgivesa n + m x [ z ]= a m x [ a n x [ z ]] : Supposea 2C t x [ z y +h.Foreachk 1setd k a=d za kanda kx[z]=P w 2h x [ z i c k w w .Wedene a k x = X w 2h x i j w j < d k a c k w w and k x [ z ]= X w 2h x [ z i j w j d k a c k w w; -3 22

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andobserve a k x [ z ]= a k x + k x [ z ] : -4 Lemma3.3.Supposea 2C t x [ z y +handd za>1.Thesequencesa kandd k ahavethefollowingproperties. i d k aiseitherstrictlyincreasingwithk,orthereisanNsuchthatifkk forall k ; iiiIfn kthenthecoecientsofa kanda nagreeonmonomialsoflengthlessthand k aand,inparticular,thecoecientsofa kanda nagreeonmonomialsoflengthlessthand k a ; iva kisaconvergentsequenceinthetopologyofC t x ygandlettinga=lim k !1 a kwehaveax=ax[ax].Moreover,aistheuniquefunctionsuchthatax=a x [ a x ] . Proof.Ifd k +1 a < 1thenthereisamonomialwappearingina k +1withj w j z >0.However,since a k +1 x [ z ]= a x [ a k x [ z ]] ,Lemma3.1iiitellsusexactly d k +1 a > d k z .Supposed n a=1forsomenandsetN=min n k : d k a = 1 o 1.Wenoted N )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 a < 1 = d N a andif k1byhypothesis,weseethatd k a d 1 a + k )]TJ/F25 11.9552 Tf 11.955 0 Td [(1 >k ,thusitemiiisproved. 23

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First,recallfromEquation3-4thata kx[z]=a kx+ kx[z].If k=0thend k a=1anda kx[ax[z]]=a kx[z],hencethecoecientsofa kanda k +1agreeuptod k a,i.e.a k = a k +1 .If k 6=0thentheminimumlengthofamonomialappearingin kx[ax[z]]isatleastd k asincetheminimumlengthofamonomialappearingin kx[z]isd k a.Hence,thecoecientsofa k +1x[z]anda kx[z]agreeonmonomialsoflengthlessthand k a,andinparticular,thecoecientsofa kanda k +1agreeonmonomialsoflengthlessthand k a.Hence,withiteration,ifn kthenthecoecientsofa k,a n,a kanda nagreeonmonomialsoflengthlessthan d k a .Thus,itemiiiisproved.Finally,toproveitemiv,weobserveda n ; a m2)]TJ/F28 7.9701 Tf 7.997 0 Td [(min f n;m grecalldisthemetriconformalpowerseries,hence a k isaCauchysequenceandthusconverges.Set a =lim k !1 a k .Letn 2 Z +begivenandnoted n a >n,byitemii.Byitemiii,thecoecientsofa,a nanda nagreeonmonomialsoflengthlessthand n a.Hence,thecoecientsofa nx,ax[a nx]anda nx[z]allagreeonmonomialsoflengthlessthann.Consequently,thecoecientsofaxandax[ax]mustagreeonallmonomialsoflengthlessthann.Thusa x = a x [ a x ] .If^ aisanyformalpowerseriesmappingsuchthatax[^ ax]=^ axthena nx[^ ax]=^ axforalln 1.However,thisimpliesthatthecoecientsofaand^ aagreeonmonomialsoflengthlessthan n ,forall n .Thus, ^ a = a . InordertoconnectitemsiiiandivtootherideasfromanalysiswedeneapartialorderingonC t x y.If=P w a w wand=P w b w w,thenwesay ifa w=0wheneverb w=0anda w=b wwhenevera w 6=0.Thus,Lemma3.3saysa nisanincreasingsequenceofpolynomialswith a asitsuniquelimit.Underthecorrectreformulation,Lemma3.3isactuallyanimplicitfunctiontheorem.InChapter4wefullydenethefreederivativeofaformalpowerseries,allowingustoeasilystateandproveTheorem4.5,theimplicitfunctiontheoremforfreeformalpowerseries. 24

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AlthoughthedenitionsandresultsinLemma3.3arevalidwheng 6=h,whenapplyingtheseideastoJacobianmatricesweoftenassume g = h . Denition3.3.Supposep 2C t x y +ghasaJacobianmatrixJ p 2 M gC t x ysuchthatJ )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 p 2 M gC t x y.Denetheauxiliaryinverseofptobe p x[z]=x J )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 pz2C t x [ z y +g andrecursivelydenethe k th auxiliaryinverse by p k x [ z ]= p x h p k )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 x [ z ] i where p 1 x [ z ]= p x [ z ] .Theindeterminatesz 1 ;:::;z g,in p x[z]are`targets'forcompositionof p withitself.Assuchitisgoodtounderstandhowtheztermsbehaveunderthesuccessivecompositions.WeimitatethesetupofLemma3.1.Forany k 1 wewrite p k x [ z ]= X w 2h x [ z i k w wwhere k w 2 C g,andforshorthandpurposeswesetd k q=d z p k.Wesplit p kintotermswithdegreelessthan d k q andthosewithdegreegreaterthanorequal d k q ; q k x = X w 2h x i j w j < d k q k w w and r k x [ z ]= X w 2h x [ z i j w j d k q k w w: Thus p k x [ z ]= q k x + r k x [ z ] : -5Sincetheminimumlengthofanymonomialappearinginr kx[ p x[z]]isgreaterthand k q,wehave k w = k +1 w forall j w j < d k q .Remark3.3.Sincetheauxiliaryinverse, p x[z]=x J )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 pz,wenotethatd 1 q=d z p >1.Hence,Lemmas3.2and3.3applyto p . Proposition3.2.Supposepisafreeformalpowerseriesmappingwithoutconstantterm.ThereisafreeformalpowerseriesmappingqwithoutconstanttermsuchthatpandqarecompositionalinversesifandonlyifJ phasafreeformalpowerseriesmultiplicativeinverse.25

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Inthiscase, J p q x = J )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 q x .Moreover, q istheuniquesolutionof q x = x J )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 p q x : -6 Proof.ByCorollary3.1weknowthatpandqarecompositionalinversesifandonlyifJ q x = J )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 p q x and J p x = J )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 q p x .Lemma3.3impliesthereexistsauniqueq 2C t x y +gsuchthat p x[qx]=qx,where p x[z]=x J )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 pzistheauxiliaryinverseofp.Sinceq 2C t x y +g,weseethatJ )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 pqx2 M gC t x yisdenedandJ pqxandJ )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 pqxareinverses.Henceq x = p x [ q x ]= x J )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 p q x and p q x = q x J p q x = x J )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 p q x J p q x = x I g = x :Next,q 2 C t x y g +alsohasanauxiliaryinverse,px[z]=x J pqz.ApplyingLemma3.3andthesameargumentasaboveweknowthereisa~ p 2 C t x y g +suchthatx = q ~ p x and ~ p x = p x [~ p x ]= x J p q ~ p x : However,since q ~ p x = x , ~ p x = x J p q ~ p x = x J p x = p x : Thus q p x = x .Therefore, q x = p x [ q x ] and p and q arecompositionalinverses. WenotethatProposition3.2doesnotrequirethatpcorrespondstoabijectivefreeanalyticmap.However,J pandJ )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 pbothexist,thuswithanapplicationofthefreeinversefunctiontheoremTheorem5in[23]wegetthatpislocallyinvertibleonsomeopenfreesetcontainingtheorigin.Wenowhaveconditionsguaranteeingaformalpowerserieshasacompositionalinverseandinfact,wehaveawaytocalculatetheinverse,oratleasttoapproximateit. 26

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Denition3.4.Weonceagainsupposez=f z 1 ;:::;z h g,wherehisnotnecessarilyequaltog.Let 2C h x [ z ih.Wesayisaproperalgebraicpolynomialifhasnoconstanttermsand d z > 1 .Wesay 2C t x y +hisasolutiontotheproperalgebraicpolynomialifx[x]= x .Each i iscalleda component ofthesolution.ByeitherLemma3.3orTheorem6.6.3in[30],everyproperalgebraicpolynomialhasauniquesolution.Let 2 C t x ywithconstanttermc.Wesayisalgebraicif )]TJ/F30 11.9552 Tf 12.353 0 Td [(cisacomponentofthesolutiontosomeproperalgebraicpolynomial.Thosefamiliarwiththisconceptwillnotethatthisdenitiondiersfromtheestablishedterminologyoftenseeninenumerativecombinatoricsandautomatatheory.Inthosecontextswedealwithsystemsofpolynomialequations,ratherthantheexplicitpolynomial:asystemofpolynomialequationsx[z]=zisaproperalgebraicsystemifisaproperalgebraicpolynomial. Proposition3.3.Supposep 2C h x i +gandJ p 2 M gC h x iistheJacobianmatrixofp.IfJ )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 p 2 M g C h x i ,thenthecompositionalinverseof p isalgebraic. Proof.Recall p ,theauxiliaryinverseofp,isgivenby p x[z]=x J )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 pz.Proposition3.2tellsusthereisaq 2C t x ygsuchthatqandparecompositionalinversesand p x[qx]=qx.Observe p x[z]hasnoconstanttermsandd z p >1.Thus p x[z]isaproperalgebraicpolynomialand q istheuniquealgebraicfunctionsatisfying p x [ q x ]= q x . WeknoweverypolynomialmappingisarationalmappingandExample6.6.5in[30]showseveryrationalmappingisanalgebraicmapping.Unfortunately,thisdoesnothelpusproveabijectivefreepolynomialhasafreepolynomialinverseatleastnotdirectly.Ifpisnotbijectivethenitmaystillhaveacompositionalinversethatisalgebraic.Theauxiliaryinversecanbeapolynomialevenifpisnotinjective,asExample3.7shows.Inthecasewherepisnotinjectivebut p isstillapolynomial,wegetauniquealgebraicfunctionqsothat p q X = X and q p X = X wheneverthesecompositionsaredened. 27

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InvertibilityoftheJacobianMatrix Inthissectionweestablishthefollowingresultaboutbijectivefreepolynomials: Theorem3.5. If p isabijectivepolynomialmapping,then J )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 p isapolynomialmatrix.ByusingtheFreeGrothendiecktheorem,wehavethateveryinjectivefreepolynomialhasafreepolynomialinverse.Hence,Theorem3.5isanunsurprisingconsequenceofthechainrule.However,Theorem3.5iscriticalfortheproofoftheFreeGrothendiecktheoremandwecannotforgoitsexposition. Lemma3.4.Ifp:MCg ! MCgisabijectivefreepolynomialandqistheinverseofpthenforeachn,thereexistsafreepolynomialr nsuchthatqX=r nXforallX 2 M n C g . Proof.ThisispartofTheoremAivandaproofcanbefoundin[23],however,forthereader'sconveniencewepresentamoredetailedargumentshowingqagreeswithafreepolynomialoneach M n C g .Let:M nCg ! C gn 2bethecanonicalisomorphism.Sincepisbijective,p[n]:M nCg ! M nCgisbijectiveandwemayviewp[n]asapolynomialingn 2variables.Thatis, p )]TJ/F28 7.9701 Tf 6.586 0 Td [(1isabijectivecommutativepolynomial,hencebytheclassicalGrothendiecktheorem, p )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 hasacommutativepolynomialinverse ^ q : C gn 2 ! C gn 2 .Sincepisabijectivefreepolynomial,qisfreeanalyticbyTheorem3.1in[12],henceq[n]:M nCg ! M nCgisanalyticandthereisapowerseries,R=P 1 m =0 P j w j = m r w wsuchthatRconvergesonM nCgandRX=q[n]XforallX 2 M nCg.Inparticular, )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 ^ q X=RX=q[n]XforallX 2 M nCg,hencedeg^ q=degq[n].Set^ R=P deg^ q m =0 P j w j = m r w wandnote^ RX=q[n]XforallX 2 M nCg.Since^ Risafreepolynomial,weconclude q agreeswithafreepolynomialon M n C g . Remark3.4.Ifpisabijectivefreepolynomialwithafreepolynomialinverseq,thenbothJ pandJ qarepolynomialmatricesandJ qpxalsoisapolynomialmatrix.ObserveI g = J q p x = J p x J q p x ,thus J )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 p 2 M g C h x i since J p x )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 = J q p x . 28

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Remark3.4isanexpectedconsequenceoftheJacobianmatrixsatisfyingthechainruleandCorollary1.4in[25]oersaslightlydierentproof.Certainlyifpisinvertiblethenitisbijective,howeverExample3.7showsthatJ p ;J )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 p 2 M gC h x iisnotsucientforp )]TJ/F28 7.9701 Tf 6.586 0 Td [(1tobeapolynomial.InthatsensethereisnoJacobianconjectureforthenoncommutativeJacobianmatrix.Ontheotherhand,Theorem3.5protsfromanoncommutativeNullstellensatzin[14]toprovetheJacobianmatrixofaninjectivefreepolynomialisinvertibleoverM gC h x i.Beforeprovingthetheorem,werststatethenoncommutativeNullstellensatzprovedbyGeorgeBergman,Theorem6.3in[14]. TheoremB.LetP C h x ibeniteandlets 2 C h x i.Letddenotethemaximumofthedegsandf deg p : p 2Pg.ThereexistsacomplexHilbertspaceHofdimensionP d j =0 g j,suchthat,if s X v =0 whenever X = X 1 ;:::;X g 2B H g , v 2H ,and p X v =0 forall p 2P ; then s isintheleftidealgeneratedby P . Theorem3.5. If p isabijectivepolynomialmapping,then J )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 p isapolynomialmatrix. Proof.Letd=max f deg p j gandsetN=g d +1.Lemma3.4tellsusq[N]agreeswithafreepolynomial.Supposes=s 1 ;:::;s gisafreepolynomialsuchthatqX=sXforallX 2 M N C g .Inparticular,each s j hasnoconstantterm.Foreach 1 j g write s j = deg s X m =1 X j w j = m j w w andobserve X j = s j p X = deg s X m =1 X j w j = m j w w p X : 29

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TakeX=X 1 ;:::;X g2 M NCgandv 2 C Nsuchthatv T p jX=0forall1 j g.Hence v T w p X =0 forall j w j > 0 since v T p j X =0 foreach 1 j g .Thus, v T X j = v T s j p X = deg s X m =1 X j w j = m j w v T w p X =0 :ByTheoremB,x jiscontainedintherightidealgeneratedbyp 1 ;:::;p g,thatis,thereexistpolynomials i;jsuchthatx j=P g i =1 p ix i;jx.LetR= i;jg i;j =1 2 M gC h x iandobserve x = p x R x = x J p x R x : Therefore J )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 p = R 2 M g C h x i . Example3.6. Let N = 0 B B @ )]TJ/F25 11.9552 Tf 9.298 0 Td [(11 )]TJ/F25 11.9552 Tf 9.298 0 Td [(11 1 C C A andset p x = x I 2 )]TJ/F30 11.9552 Tf 11.955 0 Td [(Nx 1 = x 1 ;x 2 0 B B @ 1+ x 1 )]TJ/F30 11.9552 Tf 9.298 0 Td [(x 1 x 1 1 )]TJ/F30 11.9552 Tf 11.955 0 Td [(x 1 1 C C A = x 1 + x 2 1 + x 2 x 1 ;x 2 )]TJ/F30 11.9552 Tf 11.955 0 Td [(x 2 1 )]TJ/F30 11.9552 Tf 11.955 0 Td [(x 2 x 1 : Observe, J p = I 2 )]TJ/F30 11.9552 Tf 11.955 0 Td [(Nx 1 ,andthat N 2 =0 .Hence, J )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 p = I 2 + Nx 1 and p x [ z ]= x I 2 + Nz 1 = x 1 )]TJ/F30 11.9552 Tf 11.955 0 Td [(x 1 z 1 )]TJ/F30 11.9552 Tf 11.955 0 Td [(x 2 z 1 ;x 2 + x 1 z 1 + x 2 z 1 = x 1 )]TJ/F25 11.9552 Tf 11.955 0 Td [( x 1 + x 2 z 1 ;x 2 + x 1 + x 2 z 1 : However, p isnoteveninjectiveon C 2 since p )]TJ/F25 11.9552 Tf 9.299 0 Td [(1 = 2 ; )]TJ/F25 11.9552 Tf 9.298 0 Td [(1 = 2= ; 1= p ; 1 . 30

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Note, 0 B B @ 01 00 1 C C A = 0 B B @ 01 )]TJ/F25 11.9552 Tf 9.298 0 Td [(11 1 C C A 0 B B @ )]TJ/F25 11.9552 Tf 9.299 0 Td [(11 )]TJ/F25 11.9552 Tf 9.299 0 Td [(11 1 C C A 0 B B @ 1 )]TJ/F25 11.9552 Tf 9.299 0 Td [(1 10 1 C C A ;thatis,Nissimilartoastrictlyuppertriangularnilpotentmatrix.Thus,conjugationofaJacobianmatrixbyasimilaritydoesnotpreservethedesirablepropertiesoftheJacobianmatrix.Insomesense,thenoncommutativeJacobianmatrixattemptstolinearizepolynomialmappingssothatareasonablestructureispreservedviacomposition.Infact,ifpisaformalpowerseriesmapping,thenJ pisinvertibleifandonlyifpislocallyinvertibleat0,astatementreminiscentoftheinversefunctiontheorem.Hence,J )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 pisalinearapproximationofp )]TJ/F28 7.9701 Tf 6.587 0 Td [(1at0,explainingwhywecaniterativelyconstructp )]TJ/F28 7.9701 Tf 6.586 0 Td [(1fromJ )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 p.However,Example3.7showshowtheJacobianmatrixcanfailtowitnessthenon-injectivityofapolynomial.InSection6weconstructthehypo-Jacobianmatrixofafreepolynomial,amatrixwhoseinvertibilityexactlycapturestheinjectivityornon-injectivityofthefreepolynomial. Example3.7.Thisexampleisinvestigatedin[25]anditshowsthatthereisnoJacobianconjecturewiththenoncommutativeJacobianmatrix.Letpx=x 1 ;x 2 )]TJ/F30 11.9552 Tf 12.508 0 Td [(x 1 x 2 x 1,andobserve J p x = 0 B B @ 1 )]TJ/F30 11.9552 Tf 9.298 0 Td [(x 2 x 1 01 1 C C A ;J )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 p x = 0 B B @ 1 x 2 x 1 01 1 C C A :Notepisnotbijectiveand p x[z]=x 1 ;x 2+x 1 z 2 z 1.Itisstraightforwardtoverifythat p kx[z]=x 1 ;x 2+x k 1 z 2 z 1 x k )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 1+P k )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 j =1 x j 1 x 2 x j 1andqx=x 1 ; P 1 j =0 x j 1 x 2 x j 1.Thusqiscertainlynotapolynomial.Recall p kx[z]=q kx+r nx[z].Forthisexample,q kx=x 1 ; P k )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 j =0 x j 1 x 2 x j 1whiler kx[z]=;x k 1 z 2 x k 1andd k q=2k+1.Inparticular,degq k=2k )]TJ/F25 11.9552 Tf 1.02 0 0 1 429.131 170.27 Tm [(1isstrictlyincreasingwith k ,immediatelydiscounting q frombeingapolynomial,however q isanalgebraicfunction.Infact,ifpisanyfreepolynomialwhoseauxiliaryinverse, p ,isapolynomialthentheonlywayfortheptohaveanon-polynomialinverseisifthesituationaboveoccurs,thatis,degq kisastrictlyincreasingsequence.Chapter4dealswithexactlythis. 31

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CHAPTER4 FREEDERIVATIVESANDLINEARIZATIONSInthischapterweestablishconditionsthatguaranteeq,thecompositionalinverseofp,isapolynomial.WeuseTheoremAtolinearize p ,theauxiliaryinverseofp,intermsofz 1 ;:::;z g .Thislinearizationhasthecaveatthatweintroduce g -`dummy'variables. PolynomialCriteriaWebeginbyrecallingafewfactsaboutauxiliaryinverses.ByTheorem3.5weknowifpisabijectivefreepolynomialwithp=0thenJ pandJ )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 parematricesoffreepolynomials.RecallfromDenition3.3that p k,thek thauxiliaryinverseofp,isgivenby p kx[z]=q kx+r kx[z],wheredegq k< d k q.Furthermore,byLemma3.3weknowd k q %1.Proposition3.3tellsusthatq,theinverseofp,istheuniquesolutionof p x[qx]=qx.Sinceq=lim k !1 q k,ifqwereactuallyafreepolynomialthenwewouldexpectalargedegreegaptoappearinthemonomialsof p k.ThisispreciselywhatLemma4.1dealswith. Lemma4.1.Supposeb 2C h x i +ganda 2C t x y +garecompositionalinverses,a 2C h x [ z i +gisaproperalgebraicpolynomial,andax[ax]=ax.Leta kx[z]=a kx+ kx[z]asinEquation3-3andd k a=d za kasinEquation3-1.Thefollowingareequivalent; i a isapolynomial; ii a = a m forsome m 2 Z + ; iii d N a > deg a N deg b forsome N 2 Z + . Proof.RecallfromLemma3.3thatlim k !1 a k=a,ax[ax]=axandd k aiseitheralwaysstrictlyincreasingorisstrictlyincreasinguntilitbecomesconstantatinnity.Wenotea k +1 x [ a x ]= a k x [ a x [ a x ]]= a k x [ a x ] ,implying a k x [ a x ]= a x -1 forall k 1 .Next,composingwith b x yields a k b x [ a b x ]= a b x = x . 32

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i ii .Supposeaisapolynomial.ByItemiiinLemma3.3,ifk dega,thend k a > deg a .Hence a x = a k x [ a x ]= a k x + k x [ a x ] :However,bythedenitionof k,theminimumpossiblelengthofanywordappearingin kisd k a > deg a .Thus k x [ a x ]=0 and a k = a . ii iii .Supposea=a mforsomem 2 Z +.Ifn mthenitemsiiiandivinLemma3.3implya n )]TJ/F71 11.9552 Tf 12.826 0 Td [(a mcontainsnomonomialsoflengthlessthanorequaltodega m.However,a m=a,hencewemusthavea m=a n=a,foralln m.Thus,thesequencedega ndegbisconstantforn m.Ontheotherhand,d n aiseitheralwaysstrictlyincreasingorisstrictlyincreasinguntilitbecomesconstantatinnity.Therefore,thereissomeN suchthat d N a > deg a deg b =deg a N deg b .iiii.SupposethereissomeNsuchthatd N a > dega Ndegb.Substitutingbxforx inEquation4-1, x = a N b x [ x ]= a N b x + N b x [ x ] : However, deg a N b x deg a N deg b < d N q ;andtheminimumdegreeofanymonomialappearingin Nbx[x]isgreaterthand N q,implying N b x [ x ]=0 .Thus a N b x = x ,therefore a = a N isapolynomial. Remark4.1.Supposep 2C h x i +gwithJ p ;J )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 p 2 M gC h x i.Let p betheauxiliaryinverseofpandletqbethecompositionalinverseofp.Recallthatsince p isapolynomialandd z p >1, p x[z]isaproperalgebraicpolynomialandqistheuniquealgebraicfunctionsuchthat p x [ q x ]= q x .ThusLemma4.1appliesto p and q .Itshouldbenotedthatifa=a NforsomeNthenwecannotconclude N=0.Example4.2describesabijectivefreepolynomialpwithafreepolynomialinverseq,suchthat p k x [ z ] 6 = q x ,i.e. r k x [ z ] 6 =0 forany k 1 . 33

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Example4.2.Letp 1 ;p 2 2 C h x 1 ;x 2 iwithp 1=x 1 ;x 2+x 2 1,andp 2=x 1+x 2 2 ;x 2:Botharebijectivewithq 1=x 1 ;x 2 )]TJ/F30 11.9552 Tf 13.015 0 Td [(x 2 1andq 2=x 1 )]TJ/F30 11.9552 Tf 13.014 0 Td [(x 2 2 ;x 2astheirrespectiveinverses.Theircompositionp=p 1 p 2=x 1+x 2 2 ;x 2+x 1+x 2 22hasinverseq = q 2 q 1 = x 1 )]TJ/F25 11.9552 Tf 11.955 0 Td [( x 2 )]TJ/F30 11.9552 Tf 11.955 0 Td [(x 2 1 2 ;x 2 )]TJ/F30 11.9552 Tf 11.955 0 Td [(x 2 1 : Since p 1 = x 1 x 2 1 )]TJ/F31 7.9701 Tf 6.587 0 Td [(x 1 01 )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 and p 2 = x 1 x 2 10 )]TJ/F31 7.9701 Tf 6.587 0 Td [(x 2 1 )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 ; wehave, p = x 1 x 2 10 )]TJ/F31 7.9701 Tf 6.586 0 Td [(x 2 1 )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 1 )]TJ/F28 7.9701 Tf 6.587 0 Td [( p 2 1 01 )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 = x 1 x 2 1 x 1 + x 2 2 x 2 1+ x 2 x 1 + x 2 2 ; and p x [ z ]= x 1 x 2 1 )]TJ/F28 7.9701 Tf 6.586 0 Td [( z 1 + z 2 2 01 10 )]TJ/F31 7.9701 Tf 6.586 0 Td [(z 2 1 = x 1 + x 1 z 1 + z 2 2 z 2 )]TJ/F30 11.9552 Tf 11.955 0 Td [(x 2 z 2 ;x 2 )]TJ/F30 11.9552 Tf 11.956 0 Td [(x 1 z 1 + z 2 2 :Inthiscaseagapbetweenq kandr kx[z]formsratherquicklyandthetrueinverseisextractedquiteeasily.Howevereachiterateof p k willhavea z -term. FreeDerivativesandScionsWenowintroducetheformaldirectionalderivativeaswasdonein[23]andsimilarlyin[13]and[12]. Denition4.1.Lety=f y 1 ;:::;y g gbeasetofnoncommutingindeterminatesdistinctfromxandlety=y 1 ;:::;y gbeconsideredasarowvector.Wedenethefreederivative D:C t x y ! C t x [ y ybyitsactiononmonomialsandthenextenditlinearlyandcontinuously.Dene Dx i x [ y ]= y i andrequire i D k + x [ y ]= kD x [ y ]+ D x [ y ] ; ii D x [ y ]= D x [ y ] x + x D x [ y ] , 34

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forallformalpowerseries ; 2 C t x y .Consequently,forall 2 C t x y + g wehave D x [ y ]= D x [ D 1 x [ y ] ;:::;D g x [ y ]] :ObserveDx[y]islineariny,thatis,Dx[k y+z]=kDx[y]+Dx[z].ThelinearityofthefreederivativeallowsustodeneDonmatricesofformalpowerseries.IfA 2 M m n C t x y thendene D : M m n C t x y ! M m n C t x [ y y by DA x [ y ]= DA i;j x [ y ] m;n i;j =1 : Inparticular D extendstorowvectorsintheobviousway.Remark4.3.Thederivativeinfreeanalysisisdenedbelow,andisalmostapurematrixresult.SupposeUisafreedomainhenceopenand:U! MCgisananalyticfreemap.Foranysmallenough H 2 M C g , 0 B B @ XH 0 X 1 C C A = 0 B B @ X D X [ H ] 0 X 1 C C A : -2Itturnsout,thereisastrongconnectionbetweenthefreederivativeinEquation4-2andtheformalpowerseriesderivativeinDenition4.1justifyingtheredundantuseof D .If:MCg ! MCgisafreeanalyticmapping,2C t x ygisaformalpowerseriesthatconvergesonMCg,andX=XforallX 2 MCg,thenD2C t x [ y yg convergeson M C 2 g and D X [ Y ]= D X [ Y ] forall X;Y 2 M C g .Toseethis,let=P 1 m =0 P j w j = m L w wandN=P N m =0 P j w j = m L w w.SinceN 2C h x ig,DN 2C h x [ y ig,thusDNX[Y]existsforallX;Y2 MC2 g.LetZ = XY 0 X .ByProposition6in[23], N Z = N 0 B B @ XY 0 X 1 C C A = 0 B B @ N X D N X [ Y ] 0 N X 1 C C A : 35

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ThesequenceofpolynomialsDNconvergestoDinthemetrictopologyonC t x [ y yg,thus D N X [ Y ] convergesto D X [ Y ] ,since N Z convergesto Z .Hence, Z = 0 B B @ XY 0 X 1 C C A = 0 B B @ X D X [ Y ] 0 X 1 C C A ; and D X [ Y ]= D X [ Y ] .Thus,tondthederivativeofafreeanalyticfunctionwithaformalpowerseries,itissucienttondthederivativeoftheformalpowerseriesandthenevaluatewheredesired. Example4.4.Wepresentafewformalpowerseriesandtheircorrespondingderivatives.Ifpx 1 ;x 2=x 1 x 2 )]TJ/F30 11.9552 Tf 12.522 0 Td [(x 2 x 1thenDpx 1 ;x 2[y 1 ;y 2]=y 1 x 2+x 1 y 2 )]TJ/F30 11.9552 Tf 12.522 0 Td [(y 2 x 1 )]TJ/F30 11.9552 Tf 12.521 0 Td [(x 2 y 1 :Next,ifrx 1=)]TJ/F30 11.9552 Tf 9.758 0 Td [(x 1)]TJ/F28 7.9701 Tf 6.587 0 Td [(1thenDrx 1[y 1]=)]TJ/F30 11.9552 Tf 9.758 0 Td [(x 1)]TJ/F28 7.9701 Tf 6.587 0 Td [(1 y 1)]TJ/F30 11.9552 Tf 9.758 0 Td [(x 1)]TJ/F28 7.9701 Tf 6.586 0 Td [(1.Finally,ifs 1x 1 ;x 2=x 1 ;x 2+x 2 1and s 2 x 1 ;x 2 = x 2 ;x 1 + x 2 then Ds 1 x 1 ;x 2 [ y 1 ;y 2 ]= y 1 ;y 2 + x 1 y 1 + y 1 x 1 ;Ds 2 x 1 ;x 2 [ y 1 ;y 2 ]= y 2 ;y 1 + y 2 and D s 1 s 2 x 1 ;x 2 [ y 1 ;y 2 ]= Ds 1 s 2 x 1 ;x 2 [ Ds 2 x 1 ;x 2 [ y 1 ;y 2 ]] = y 2 ; y 1 + y 2 + x 2 y 2 + y 2 x 2 :Beforeproceedingwithourinvestigationofthefreederivative,westoptoquicklyprovetheimplicitfunctiontheoremforncformalpowerseries.Forananalyticapproachtotheimplicitfunctiontheoremfor M C g see[1]. Denition4.2. Suppose z = f z 1 ;:::;z h g and f x ; z 2 C t x [ z y h .Dene @f @ z = Df i x ; z [0 ;e j ] h i;j =1 2 M h C t x [ z y ; where e j isthestandardvectorwitha 1 inthe i th positionand 0 elsewhere. 36

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Theorem4.5Implicitfunctiontheorem.Supposefx ; z2C t x [ z yh.Iff;0=0and@f=@ z;02 M hCisinvertible,thenthereexistsauniqueg 2C t x yhsuchthatg =0 and f x ; g x =0 . Proof.Sincef;0=0,weseethatfx ; zhasnoconstantterms.Bycomposingwithanappropriatechangeofvariables,wemayassume@f=@ z;0=I h.Hence,thecoecientofeachz iterminf jis i;j,theKroneckerdelta.Set^ fx[z]=z )]TJ/F30 11.9552 Tf 11.996 0 Td [(fx ; zandnote^ fsatisestheconditionsofLemma3.3.Thus,thereexistsauniqueg 2C t x yhsuchthatg=0and^ f x [ g x ]= g x .Finally,since f x ; z = z )]TJ/F25 11.9552 Tf 14.503 3.155 Td [(^ f x [ z ] , f x ; g x = g x )]TJ/F25 11.9552 Tf 14.503 3.155 Td [(^ f x [ g x ]= g x )]TJ/F71 11.9552 Tf 11.956 0 Td [(g x =0 ; -3andtheuniquenessofgfor^ fimpliesgistheuniqueformalpowerseriessatisfyingbothg =0 andEquation4-3. Denition4.3.Supposep 2C h x ig.Wedenethescionofp,F 2C h x [ y i2 g,byFx ; y=Dpy[x]; y.Furthermore,ifweviewpasafreepolynomialfromMCgtoM C g ,then F : M C 2 g ! M C 2 g isafreepolynomialand F X;Y = Dp Y [ X ] ;Y .OfparticularimportanceisthefactthatFisx-linear,thatis,Fx+z ; y=Fx ; y+F z ; y .Moreover, DF x ; y [ z ; w ] isautomatically z -linear. Proposition4.1.Supposepisafreepolynomial.IfFisthescionofp,thenpisbijectiveifandonlyif F isbijective. Proof.SupposeJ pistheJacobianmatrixofp.Sincepx=x J px,theJacobianmatrixofF isgivenby J F x ; y = 0 B B @ J p y 0 DJ p y [ x ] I 1 C C A : 37

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WenotethatFx ; y=x ; yJ Fx ; y.Leth=h 1 ;:::;h gandk=k 1 ;:::;k gbeg -tuplesofnoncommutingindeterminatestreatedasrowvectors.Consider DF x ; y [ h ; 0] = h J p y + x DJ p y [0]+0 DJ p y [ x ]+ y D DJ p y [ x ][ h ; 0] ; 0 = h J p y +0+0+ y DJ p y [ h ] ; 0 = Dp y [ h ] ; 0 :ThemotivationforwhyDDJ py[x][h ;0]=DJ py[h]hingesonthex-linearityofDJ p y [ x ] .Toclarifythispointwedemonstrateonamonomial m x ; y = y x i y : Dm x ; y [ h ; 0]= D y [0] x i y + y h i y + y x i D y [0] = m h ; y :RecallF g + ix ; y=y g + ifor1 i g,henceDF g + ix ; y[h ; k]=k i.Inparticular,ifX;Y;H;K 2 M nCgthenDFX;Y[H;K]=0impliesK=0.ThusDpY[H]=0ifandonlyifDFX;Y[H;K]=0.Therefore,anapplicationofTheoremAimpliespisbijectiveifandonlyif F isbijective. Lemma4.2.Supposepisabijectivefreepolynomialwithnoconstantterm,Fisthescionofp,andqandGarethecompositionalinversesofpandFrespectively.LetGx ; y[z ; w]betheauxiliaryinverseof F .Then, i G x ; y [ z ; y ] satisestheconditionsofLemma3.3with G x ; y asitsuniquesolution; ii G x ; y [ z ; y ] isane z -linear; iii G x ; y = Dq p y [ x ] ; y ; ivif q isafreepolynomialthen G isafreepolynomialand deg q deg G deg p deg q : 38

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Proof.SinceGisthecompositionalinverseofF,itautomaticallysatisestheconditionsofLemma3.3.Hence, G x ; y [ z ; w ] has G x ; y asitsuniquesolution.RecallF g + ix ; y=y ifor1 i g.ItfollowsthatG g + ix ; y[z ; w]=y iandthusG g + ix ; y=y i.Inparticular,Gx ; y[z ; y]stillsatisestheconditionsofLemma3.3andGx ; yistheuniquesolutionofGx ; y[z ; y].Thus,Lemma4.1appliestoGx ; y[z ; y]and G x ; y ,justifyingouruseof G x ; y [ z ; y ] inlieuof G x ; y [ z ; w ] .Toproveitemii,letJ pandJ FbetheJacobianmatricesofpandF,respectively.SincebothpandFhavecompositionalinverses,Proposition3.2saysJ )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 pandJ )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 Fexistasmatricesofformalpowerseries.Hence, J )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 F z ; w = 0 B B @ J p w 0 DJ p w [ z ] I 1 C C A )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 = 0 B B @ J p w )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 0 )]TJ/F30 11.9552 Tf 9.298 0 Td [(DJ p w [ z ] J p w )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 I 1 C C A :Sincezonlyappearsinafreederivative,weseethatJ )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 Fz ; wisanez-linear.Thus,G x ; y [ z ; y ]= x ; y J F z ; y mustalsobeane z -linear. Foriii,since q p x = x ,wehave x = D q p y [ x ]= Dq p y [ Dp y [ x ]] .Thus, x ; y = Dq p y [ Dp y [ x ]] ; y = Dq p y [ F x ; y ] ; y : -4 Since G g + i x ; y = y i ,substituting G x ; y for x ; y intoEquation4-4yields G x ; y = Dq p y [ F G x ; y ] ; y = Dq p y [ x ] ; y :Lastly,supposeqisafreepolynomial.ItfollowsthatDqisafreepolynomial,henceGisafreepolynomial.Since deg q deg Dq p y [ x ] deg q deg p weconclude deg q deg G deg q deg p : Inatrivialsense,thedegreeboundsbetweenGandqarenotstrict.Ifpx=xthenq x = x , F x ; y = x ; y and G x ; y = x ; y .Hence, deg q =deg q deg p =deg G . 39

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Remark4.6.Weemphasizeapointmadeintheproofof4.2;sinceGisthesolutiontobothG x ; y [ z ; y ] and G x ; y [ z ; w ] wemayuse G x ; y [ z ; y ] ratherthan G x ; y [ z ; w ] . Example4.7. Let p x = x 1 ;x 2 )]TJ/F30 11.9552 Tf 11.955 0 Td [(x 2 1 and F x ; y = x 1 ;x 2 )]TJ/F30 11.9552 Tf 11.955 0 Td [(x 1 y 1 )]TJ/F30 11.9552 Tf 11.955 0 Td [(y 1 x 1 ;y 1 ;y 2 : Hence p x [ z ]= x 1 ;x 2 + x 1 z 1 and D p x ; z [ h ; k ]= h 1 ;h 2 + h 1 z 1 + x 1 k 1 andGx ; y[z ; y]=D p py; y[x ; z]; y=x 1 ;x 2+x 1 y 1+x 1 z 1 ;y 1 ;y 2.Note p 2x[z]=x 1 ;x 2+x 2 1=qxandG 2x ; y[z ; y]=x 1 ;x 2+x 1 y 1+x 1 x 1 ;y 1 ;y 2.Lastly,2=deg q =deg G ,while deg q deg p =4 ,so deg q =deg G < deg q deg p .Proposition4.1tellsusthatapolynomial,p,isbijectiveifandonlyifitsscion,F,isbijective.Thescionisx-anelinear,anditsinversefunction,G,istheuniquealgebraicsolutiontoaproperalgebraicpolynomialthatisz-anelinear.Weinvestigatepreciselytheformalpowerseriesthataregeneratedbysuchz-anelinearproperalgebraicpolynomialsinChapter6. 40

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CHAPTER5 DEGREEBOUNDSONNCRATIONALMAPSInordertoproveTheorem6.7werequireresultsabouthowrationalfunctionsbehavewhenevaluatedonmatrices.Usingrationaldegreesonncrationalfunctions,weproveProposition5.1,aresultaboutthebehaviorofncrationalfunctionswhentheyareevaluatedongenericmatrices. RationalDegreeBoundsInthissectionweintroducetopicsfromnoncommutativealgebrainordertoproveageneralprinciple;evaluatinganoncommutativerationalfunctionronatupleofmatricesproducesamatrixwhoseentriesbehavesimilarlytor.Amajorobstacleinprovingthisprincipleisthefactthatnoncommutativerationalfunctionscannotalwaysbewrittenasafractionofpolynomials.However,byintroducingacommutingindeterminatetweareabletocharacterizethedegreeofancrationalfunctionanditsevaluationsonmatrices. Denition5.1.SupposeUisaskeweldcontainingCandU[t]isthepolynomialringoverU.Wedenethemapdeg t:U[t]! Z [fginthenaturalway;deg t=andifr = r 0 + r 1 t + + r m t m , r m 6 =0 ,then deg t r = m .Forany r;s 2 U [ t ] , i deg t rs =deg t r +deg t s ; ii deg t x + y max f deg t x ; deg t y g .Theorem2.1.15in[22]tellsusU[t]=U C[t]isanOredomainwithaclassicalringofquotients, U t .Hence,forany r 2 U t thereexist ; 2 U [ t ] suchthat r = )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 .Wecanuniquelyextenddeg ttoUtbyTheorem9.1in[4]suchthatdeg tr )]TJ/F28 7.9701 Tf 6.587 0 Td [(1=)]TJ/F25 11.9552 Tf 11.291 0 Td [(deg tr,forallr 6=0.Inparticular,ifr= )]TJ/F28 7.9701 Tf 6.586 0 Td [(1thendeg tr=deg t)]TJ/F25 11.9552 Tf 12.01 0 Td [(deg t.Wesaydeg t : U t ! Z [f1g isa rationaldegreemap .Note:itemiistrueaslongasUisadomainUhasnozerodivisors.IfUhaszerodivisorsthen deg t rs deg t r +deg t s . ThemainresultofSection5isasfollows. 41

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Proposition5.1.Supposew=f w 1 ;:::;w h gisacollectionoffreelynoncommutingindeterminates,w=w 1 ;:::;w h,tisacentralindeterminateandN 2 Z +.Letr 2 C < w >beanonzerorationalfunctionsuchthatrt w=w[t]w[t])]TJ/F28 7.9701 Tf 6.586 0 Td [(1,where; 2 C < w >[t].Suppose i S isaeldcontaining C , ii U M N S isaskeweldgeneratedby u 1 ;:::;u h 2 M N S ,each u i 6 =0 . If deg t istherationaldegreemapon S t ,then deg t r tu 1 ;:::;tu h i;j deg t w [ t ] 2 Z ; forall 1 i;j N ,whenever r u 1 ;:::;u h isdened.OfparticularimportanceinProposition5.1isthatdeg tw[t]isindependentofN,henceitappliesquitenicelytofreefunctions.Remark5.1.Thefollowingdenitionswillbefamiliartoanalgebraistbutperhapsnottoananalyst.IfRisanycommutativeintegraldomain,thentheeldoffractionsofRisthesmallesteldinwhich R canbeembedded.Everyintegraldomainhasaeldoffractions.Next,aringDissaidtobeanoncommutativedomainifithasnozerodivisors,i.e.ifa;b 2 Dsuchthatab=0theneithera=0orb=0.If,inaddition,everynonzeroelementofD hasamultiplicativeinversethen D issaidtobeaskeweld.LetRbeanoncommutativedomainandletSbethesetofallthenonzeroelementsofR.WesayRisarightOredomainifforeveryr 2 Rands 2 S,rS sR 6=?.IfRisarightOredomain,thenthereisauniqueuptoR-isomorphismskeweldDcontainingRasasubringsuchthateveryelementofDhastheformrs )]TJ/F28 7.9701 Tf 6.586 0 Td [(1,fors;r 2 Rands 6=0.Inthiscase,theskeweld D iscalledtheclassicalringofquotientsof R ,anditisuniqueuptoisomorphism. 42

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Lemma5.1.SupposeUisanyskeweld,v 2 U[t]nisarowvectorofpolynomialsandM 2 M n U t isan n n matrix.If deg t isarationaldegreemapon U t ,then deg t vM i max k f deg t v k g +max k f deg t M k;i g ; foreach 1 i n . Moreover,if deg t v i and deg t M i;j for 1 i;j n then max 1 k n n deg t vM k o + : Proof. Fix 1 i g .Simplyapplyingthepropertiesofthedegreemap, deg t vM i =deg t X k v k M k;i ! max k f deg t v k M k;i g =max k f deg t v k +deg t M k;i g max k f deg t v k g +max k f deg t M k;i g :Next,supposedeg tv i anddeg tM i;jforall1 i;j n.Thus,max k f deg t v k g andmax i;k f deg t M k;i g.Finally,sincedeg tvMi +for1 i n ,weconclude max i f deg t vM i g + . Letw=f w 1 ;:::;w h gbeanitecollectionoffreelynoncommutingindeterminatesandletw=w 1 ;:::;w h.WerecallafewfactsabouttheconstructionofC < w >,thealgebraofnoncommutativerationalfunctions.Theseresultsanddenitionscanbefoundin[19]and[20].LetR CwbethesetofallnoncommutativerationalexpressionsoverC,i.e.allpossiblesyntacticallyvalidcombinationsofelementsinCandw,arithmeticoperationsaddition,multiplication,inversionandparentheses.Forexample,w 1+w 1,w 1w 2 )]TJ/F30 11.9552 Tf 11.993 0 Td [(w 1)]TJ/F28 7.9701 Tf 6.587 0 Td [(1and0)]TJ/F28 7.9701 Tf 6.587 0 Td [(1aresyntacticallyvalidcombinations.Theinversionheightof 2R Cwisthemaximumnumberofnestedinversesin .ThesubsetofMChatwhichisdenedisdenoteddom andiscalledthedomainof.Wesay 2R CwisnondegenerateifAisdenedforsomeA 2 MCh.If 1 ; 2are43

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nondegeneraterationalexpressionsthenwesay 1 2ifandonlyif 1A= 2AforallA 2 dom 1 dom 2.ThisrelationisanequivalencerelationonthesetofallnondegeneraterationalexpressioninR Cw.WedeneC < w >,theskeweldofnoncommutativerational functions,tobethesetofequivalenceclassesofnondegenerateexpressionwithrespectto.Ifr 2 C < w >,thenthedomainofr,denoteddom r,isdenedastheunionofthedomainsofallrepresentativesofrandifA 2 dom rthenrA=Aforanyrepresentative 2R Cwsuchthat A 2 dom .Remark5.2.BothC h w iandC rat t w yembedintoC < w >,andinfact,ifr 2 C isdenedat0thenr 2 C t w y.SinceeveryrationalseriesseeDenition2.1isdenedat0,wehavethattherationalseriesareexactlythencrationalfunctionsdenedat 0 . WenowintroducealemmathatwillbeimplicitlyusedthroughouttherestofChapter5. Lemma5.2. Suppose r 2 C < w > .If t isacentralindeterminate,then r t w 2 C < w > t . Proof. Theprooffollowsquicklyfrominductionontheinversionheight. Example5.3.Ifisarationalfunctionincommutingvariablesthencanbewrittenasafractionofpolynomials;=pq )]TJ/F28 7.9701 Tf 6.587 0 Td [(1.Hence,itmakessensetotalkaboutarationaldegreemap,deg =deg p )]TJ/F25 11.9552 Tf 11.955 0 Td [(deg q .Inthenoncommutativecasewecannotguaranteethatarationalfunctionrcanbewrittenasafractionofpolynomials.However,thecommutingindeterminatetweintroduceactsasayardstickfortherationaldegreeofr.Sincedeg tqt)]TJ/F28 7.9701 Tf 6.587 0 Td [(1=)]TJ/F25 11.9552 Tf 11.291 0 Td [(deg tqtwecanunpackarationalfunctionbymovingiterativelythroughtheinversionheights. Forexample,if r x 1 ;x 2 = x 1 )]TJ/F30 11.9552 Tf 11.955 0 Td [(x 1 )]TJ/F30 11.9552 Tf 11.955 0 Td [(x 2 )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 x 1 )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 44

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thenwecanguessdegx 1)]TJ/F30 11.9552 Tf 12.16 0 Td [(x 2)]TJ/F28 7.9701 Tf 6.587 0 Td [(1 x 1=2)]TJ/F25 11.9552 Tf 1.02 0 0 1 306.982 708.045 Tm [(1,deg)]TJ/F30 11.9552 Tf 12.16 0 Td [(x 1)]TJ/F30 11.9552 Tf 12.161 0 Td [(x 2)]TJ/F28 7.9701 Tf 6.586 0 Td [(1 x 1)]TJ/F28 7.9701 Tf 6.586 0 Td [(1=1)]TJ/F25 11.9552 Tf 1.02 0 0 1 502.473 708.045 Tm [(2and,deg x 1 )]TJ/F30 11.9552 Tf 11.956 0 Td [(x 1 )]TJ/F30 11.9552 Tf 11.955 0 Td [(x 2 )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 x 1 )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 =2 )]TJ/F25 11.9552 Tf 11.955 0 Td [(2 .Introducingthecommuting t weget r tx 1 ;tx 2 = tx 1 )]TJ/F30 11.9552 Tf 11.955 0 Td [(tx 1 )]TJ/F30 11.9552 Tf 11.955 0 Td [(tx 2 )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 tx 1 )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 = tx 1 x )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 1 x )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 1 )]TJ/F30 11.9552 Tf 11.955 0 Td [(t 2 x 1 )]TJ/F30 11.9552 Tf 11.955 0 Td [(tx 2 )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 = t )]TJ/F30 11.9552 Tf 11.955 0 Td [(tx 2 x )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 1 )]TJ/F30 11.9552 Tf 11.955 0 Td [(tx 2 )]TJ/F30 11.9552 Tf 11.956 0 Td [(t 2 x 1 )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 = t )]TJ/F30 11.9552 Tf 11.955 0 Td [(t 2 x 2 x )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 1 )]TJ/F30 11.9552 Tf 11.955 0 Td [(tx )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 1 x 2 )]TJ/F30 11.9552 Tf 11.955 0 Td [(t 2 x 1 )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 ; afractionofpolynomialsin t withcoecientsin C < x > .Infact,asweguessed, deg t r t x = 2 )]TJ/F25 11.9552 Tf 11.955 0 Td [(2=0 . WearenowinapositiontoproveProposition5.1. Proposition5.1.Supposew=f w 1 ;:::;w h gisacollectionoffreelynoncommutingindeterminates,w=w 1 ;:::;w h,tisacentralindeterminateandN 2 Z +.Letr 2 C < w >beanonzerorationalfunctionsuchthatrt w=w[t]w[t])]TJ/F28 7.9701 Tf 6.586 0 Td [(1,where; 2 C < w >[t].Suppose i S isaeldcontaining C , ii U M N S isaskeweldgeneratedby u 1 ;:::;u h 2 M N S ,each u i 6 =0 . If deg t istherationaldegreemapon S t ,then deg t r tu 1 ;:::;tu h i;j deg t w [ t ] 2 Z ; forall 1 i;j N ,whenever r u 1 ;:::;u h isdened. Proof.Supposeu=u 1 ;:::;u hand 2 U[t]isanonzeropolynomialsuchthatu[t]=P m j =1 k j u t j with k m u 6 =0 .Note deg t u [ t ]=max 1 i;j N f deg t u [ t ] i;j g = m: Werstshowthat deg t det u [ t ]= Nm . 45

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Fromthedenitionofthedeterminant,deg tdetu[t] Nm.Infact,detu[t]isasumofproductsofNentriesofu[t].Hence,detu[t]2 S[t]anditst Nmcoecientisexactlydetk mu.SinceSisaeld,foranyA 2 M NS,detA6=0ifandonlyifAisinvertible.Next,Ubeingaskeweldandk m 2 Ubeingnonzeroimplyk misinvertible,hencedet k m 6 =0 .Thus, deg t det u [ t ]= Nm .Thatis, deg t det u [ t ]= N deg t u [ t ] : -1Next,werecallthattheadjugateofanyN Nmatrixisamatrixofdeterminantsof N )]TJ/F25 11.9552 Tf 11.955 0 Td [(1 N )]TJ/F25 11.9552 Tf 11.955 0 Td [(1 sub-matrices.Hence,forall 1 i;j N , deg t adj u [ t ] i;j N )]TJ/F25 11.9552 Tf 11.955 0 Td [(1deg t u [ t ] : -2 Since u [ t ] )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 =det u [ t ] )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 adj u [ t ] ,Equation5-1andInequality5-2imply deg t u [ t ] )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 =max i;j deg t adj u [ t ] i;j det u [ t ] ! 0 : -3Next,rt w=w[t]w[t])]TJ/F28 7.9701 Tf 6.586 0 Td [(1forsome; 2 C < w >[t]andassumingruisdened,r t u = u [ t ] u [ t ] )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 .Hence, deg t r t u i;j =deg t N X ` =1 u [ t ] i;` u [ t ] )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 `;j ! max ` deg t u [ t ] i;` u [ t ] )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 `;j max ` f deg t u [ t ] i;` g +max ` deg t u [ t ] )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 `;j deg t u [ t ]+deg t u [ t ] )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 deg t u [ t ] : WherethelastinequalityusesEquation5-3.Finally, deg t u [ t ] deg t w [ t ] implies deg t r t u i;j deg t u [ t ] deg t w [ t ] forall 1 i;j N . 46

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ItshouldbeemphasizedthatEquation5-1isnottrueforanypolynomialinM NS[t].Rather,itholdstrueifandonlyiftheleadingcoecientisaninvertiblematrix.Forexample,if t = 0 B B @ 10 01 1 C C A + 0 B B @ 10 00 1 C C A t 2 + 0 B B @ 01 10 1 C C A t 3 + 0 B B @ 00 01 1 C C A t 4 = 0 B B @ 1+ t 2 t 3 t 3 1+ t 4 1 C C A then deg t t =4 while deg t det t =deg t + t 2 + t 4 =4 < 8 .Ontheotherhand,if t = 0 B B @ 10 01 1 C C A + 0 B B @ 01 10 1 C C A t 2 + 0 B B @ 10 01 1 C C A t 4 = 0 B B @ 1+ t 4 t 2 t 2 1+ t 4 1 C C A then deg t t =4 and deg t det t =deg t + t 4 + t 8 =8 .Proposition5.1givescredencetothenotionthatifrisanoncommutativerationalfunction,thenrsisamatrixofrationalfunctionswhosebehaviorismodeledbyr.Inparticularwewillapplythisideatogenericmatrixalgebras. GenericMatrixAlgebrasSupposen 2 Z +.Foreachi 2 Z +,1 j g,and1 k;` n,let i ;j n;k;`beacommutingindeterminate.Next,foreachi 2 Z +,set i n=f i ;j n;k;`:1 j g;1 k;` n g.Ifi; ^ { 2 Z +,thenthealgebrasC[ i n]andC[ i n [ ^ { n]haveeldsoffractionsC i nandC i n [ ^ { n ,respectively.Fori;n 2 Z +and1 j g,dene i ;j n= i ;j n;k;`n k;` =1 2 M nC[ i n]tobeageneric matrixofsizen.DeneGM n i tobethealgebraofgenericmatrices;thatis,theunitalC -subalgebraof M n C [ i n ] generatedby i ; 1 n ;:::; i ;g n .Let i n = i ; 1 n ;:::; i ;g n bea g -tupleofgenericmatrices.TopreparefortheiruseinChapter6,wedeneGM n i Ttobethealgebraoftransposedgenericmatrices,thatis,GM n i Tisthealgebrageneratedbythetransposed47

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genericmatrices i ; 1 n T ;:::; i ;g n T .Let i n T = i ; 1 n T ;:::; i ;g n T bea g -tupleoftransposedgenericmatricesandlet T; j n denotethe 2 g -tuple j ; 1 n T I n ;:::; j ;g n T I n ;I n j ; 1 n ;:::;I n j ;g n where istheKroneckerproduct.Lastly,let GM n T; j = GM n j T GM n j andobserveGM n T; j isgeneratedby f i n T I n g[f I n j n g .Remark5.4.ByLemma2.5andProposition2.6in[19],GM n i GM n j iscontainedinaskeweld,UD n i ; j .SinceGM n i GM n j andGM nT; j areisomorphicasalgebras,GM n T; j mustbecontainedinsomeskeweld,UD n T; j . Thus,forany n 2 Z + , i C n isaeld, iiGM n T; M n 2 C n isthe C -algebrageneratedby f n T I n g[f I n n g , iiiGM n T; UD n T; M n 2 C n andUD n T; isaskeweld.Hence,Proposition5.1isapplicable;ifr 2 C < y [ x >,thenthereexistsad r 2 Z +suchthatdeg t r t T; n i;j d r ,forall n 2 Z + and 1 i;j n 2 ,whenever r T; n isdened. Example5.5.BytheCayley-Hamiltontheorem,anyX 2 M 2CsatisestherelationX 2=c 1 X+c 0 I 2,forsomescalarsc 1 ;c 0 2 C.TakethecommutatorofbothsidesagainstY 2 M 2 C , [ X 2 ;Y ]= c 1 [ X;Y ]+ c 0 [ I 2 ;Y ]= c 1 [ X;Y ] : Next,takethecommutatorofbothsidesagainst [ X;Y ] , [[ X 2 ;Y ] ; [ X;Y ]]= c 1 [[ X;Y ] ; [ X;Y ]]=0 : 48

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Thus,py;x=[[x 2 ;y];[x;y]]vanishesonM 2C,i.e.pisapolynomialidentityforM 2C.Letry;x=x 2)]TJ/F30 11.9552 Tf 11.996 0 Td [(py;x)]TJ/F28 7.9701 Tf 6.587 0 Td [(1andnoterisafractionofthepolynomialsx 2and1)]TJ/F30 11.9552 Tf 11.996 0 Td [(py;x.Thus,Proposition5.1impliesdeg trt T; ni;j deg tt 2 x 2=2,foralln 2 Z +.Inparticular,if n< 3 then p t T; n = I 2 and deg t r t T; n i;j =2 . Lemma5.3.Suppose:MCg ! MCgisafreeanalyticfunctionwithapowerseriesthatconvergesforeach X 2 M C g and n isapolynomialforeach n .Dene T = n deg t j t n k;` : n 2 Z + ; 1 j g; 1 k;` n o : If T isbounded,then isafreepolynomial. Proof. ThisisProposition3.1in[18]. 49

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CHAPTER6 HYPORATIONALSERIESProposition4.1showsthatifpisafreepolynomial,thenFx ; y=Dpy[x]; yisafreepolynomial,andpisinjectiveifandonlyifFisinjective.Lemma4.2impliesG,theinverseofF,istheuniquesolutiontoaproperalgebraicpolynomialthatisz-anelinear.Thez-anelinearityisreminiscentofrealizationsofncrationalfunctions,see[34]and[20].Withthissimilaritytorealizationsinmind,wegeneralizetheclassofrationalseriesseeDenition2.1toaslightlylargerclassofformalpowerseriesthatwecallthehyporationalseries.Inparticular,thescionofafreepolynomialhasahyporationalseriesasitscompositionalinverse.WeshowthateveryrationalseriesishyporationalandTheorem6.7,themainresultofSection6,saysthatahyporationalserieswithoutsingularitiesisafreepolynomial.InSection6,weapplythesametechniquesusedtoanalyzehyporationalseriestothefreederivativeofpolynomials.Thisleadstotheconstructionofthehypo-JacobianmatrixofafreepolynomialmappingandTheorem6.9.Finally,combiningTheorem6.9withresultsonautomorphismsof C h x i provesthemainresultofthispaper,Theorem6.11. HyporealizationsandHyporationalSeries Denition6.1.Oncemore,supposez=f z 1 ;:::;z h gisasetoffreelynoncommutingindeterminateswherehisnotnecessarilyequaltog.Leta 2C h x [ z ih.RecallfromDenition3.4thataisaproperalgebraicpolynomialifahasnoconstanttermandd za>1,thatis,ifwisamonomialappearinginawithj w j z >0thenj w j2.Wesayaisz-anelinearifax[z]=ax+x[z],wherex[z+w]=x[z]+x[w].Ifaisbothaproperalgebraicpolynomialand z -anelinearthenwesay a isa hyporealization .Supposer 2 C t x ywithconstanttermr 1.Wedenertobeahyporationalseriesifthereexistsahyporealizationasuchthatr )]TJ/F30 11.9552 Tf 11.972 0 Td [(r 1isacomponentofthesolutionofa.Namely,thereexistsr=r 1 ;:::; r h2C t x y +hsuchthatax[r]=randr )]TJ/F30 11.9552 Tf 12.648 0 Td [(r 1=r i.LetC hyp t x y C t x y denotethecollectionofallhyporationalseries. RecallfromDenition2.1that C rat t x y isthealgebraofrationalseries. 50

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Remark6.1.Werecallseveralfactsfromrealizationtheory.LetC < x > 0 C < x >denotethesubringofncrationalfunctionsthatareregularattheorigin: C < x > 0 = f r 2 C < x > :0 2 dom r g :AswasmentionedinRemark5.2,C < x > 0=C rat t x y.Ifr 2 C < x > 0,thenrhasarealization;thereexist d 2 Z + , c;b 2 M d 1 C and A 1 ;:::;A g 2 M d C suchthat r = c T 0 @ I d )]TJ/F31 7.9701 Tf 17.149 13.615 Td [(g X j =1 A j x j 1 A )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 b:Classicalrealizationtheoryhasalongandstoriedhistoryinbothmathematicsandappliedelds.Weusedenitionsandresultsfrom[20],whichprovidesanexcellentexpositionofrealizationsofncrationalfunctionsandtheirdomains.Remark6.2.Everyrationalseriesishyporational,thatis,C rat t x y C hyp t x y.Weomittheproofoftheabovestatement,howeveritfollowsreadilyfromarearrangementoftherealizationofagivenncrationalmap.Infact,ifrisaformalpowerserieswithconstanttermr 1,thenr 2 C ratt x yifandonlyifthereexistA 2 M hC h x i +anda 2C h x i +h,suchthatr )]TJ/F30 11.9552 Tf 12.405 0 Td [(r 1isacomponentofthesolutiontothehyporealizationax[z]=ax+z Ax.Thisconditionpreciselydelineatesthedierencebetweenrationalseriesandhyporationalseriesthatarenotrational.Realizationtheorytellsusthatthereisaveryintimaterelationshipbetweenrationalfunctionsandlinearity.Example3.7providesuswithafunctionthatishyporationalbutnotrational. Example6.3.Thehyporealizationsx[z]=x 2+x 1 z 2 x 1hasthesolutionsx=P 1 n =0 x n 1 x 2 x n 1.Thus s ishyporationaland s x [ z ] isahyporealizationof s .Arguingbycontradiction,supposesisrational.Henceshasaminimalrepresentationsx=c TI )]TJ/F30 11.9552 Tf 13.269 0 Td [(Ax 1 )]TJ/F30 11.9552 Tf 13.269 0 Td [(Bx 2)]TJ/F28 7.9701 Tf 6.586 0 Td [(1 b,whereA;B 2 M nCandb;c 2 M n 1Cg.Letm At=a 0+a 1 t++a d t dbetheminimalpolynomialofAandnotethereisaksothat51

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a k 6 =0 since A 6 =0 .Observe c T A i BA j b = i;j where istheKroneckerdelta.Hence 0= c T 0 b = c T a k m A A BA k b = d X i =0 a i a k c T A i BA k b = j a k j 2 c T A k BA k b = j a k j 2 > 0 ; acontradiction.Therefore, s isnotrational.Oneofthemainadvantagesoftherealizationtheoryofncrationalfunctionsisthattheintrinsiclinearityofrationalsisexpressedthroughmatrices.Sincehyporationalseriesaregeneratedbylinearproperalgebraicpolynomials,wewouldliketoimitaterationalrealizationtheoryforhyporationalseries.ThisispreciselywhatwedoinSection6. HypomatrixRepresentations Denition6.2.ForanyC-algebraRandn 2 Z +,themapvec[n]:M nR! M 1 n 2Rgivenby vec [ n ] A i )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 n + j = A i;jisalinearisomorphismtakingann nmatrixwithentriesinRtoalengthn 2rowvectorwithentriesin R .IfU;V 2 M nRthenU V 2 M n 2RisthestandardKroneckerproduct.Furthermore,ifZ;U;V 2 M nRthentheproductvec[n]ZU Vistheproductofa1 n 2rowvectorandan n 2 n 2 matrix. Letussee vec [ n ] inaction.If A 2 M 3 C [ x 1 ;:::;x 9 ] with A = 0 B B B B B B @ x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 1 C C C C C C A ; then vec [3] A = x 1 ;x 2 ;x 3 ;x 4 ;x 5 ;x 6 ;x 7 ;x 8 ;x 9 . 52

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Fornotationalconvenience,weallow vec [ n ] toapplycoordinate-wisetotuples: vec [ n ] A 1 ;:::;A m = vec [ n ] A 1 ;:::; vec [ n ] A m : Lemma6.1. If U;V;Z 2 M n R then vec [ n ] UZV = vec [ n ] Z U T V 2 M 1 n 2 R : Proof.Ourdenitionofvec[n]isaleftsidedversionofthevecfunctiondenedat4.2.9in[15].ByadaptingLemma4.3.1in[15]weconcludevec[n]UZV=vec[n]ZU T V. Recall n = ; 1 n ;:::; ;g n isa g -tupleofgenericmatriceswhile T; n = ; 1 n T I n ;:::; ;g n T I n ;I n ; 1 n ;:::;I n ;g n ; isa 2 g -tupleof n 2 -matricesover C [ i ] . Borrowingfrom[19],wedene C h y x i . . = C h y i C h x i = C h y [ x i . [ y i ;x j ]:1 i;j g ;tobethebipartitefree C -algebra.ThealgebraC h y x iiscontainedinaskeweldoffractions, C < y x > ,thebipartiterationalfunctions.Remark6.4.Webrieydenethetransposeofapolynomial.Foranyw 2h x iwithw=x i 1 x i 2 :::x i n,wesayw T=x i n x i n )]TJ/F29 5.9776 Tf 5.756 0 Td [(1 :::x i 1.Hence,foranypolynomialp=P p w w,wedenep T=P p w w T.Inparticular,ifX=X 1 ;:::;X gisatupleofmatrices,thenp T X T = p X T . 53

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Proposition6.1.Supposea 2C h x [ z igisahyporealizationwithax[z]=ax+x[z].Ifa 2C t x y +gisthehyporationalseriessuchthatax[ax]=ax,thenthereexists 2 M g C h y x i suchthat i vec[n] a X [ Z ]=vec[n]aX+vec[n]ZX T I;I X,forallnandX;Z 2 M n C g ; ii I )]TJ/F25 11.9552 Tf 11.955 0 Td [( y ; x isinvertibleasamatrixover C < y x > ; iii dom nm I )]TJ/F25 11.9552 Tf 11.955 0 Td [( )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 i;j 6 = ? forall n;m 2 Z + and 1 i;j g ; iv I )]TJ/F25 11.9552 Tf 11.955 0 Td [( T; n )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 isdenedforall n 2 Z + ; v vec [ n ] a n = vec [ n ] a n I )]TJ/F25 11.9552 Tf 11.955 0 Td [( T; n )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 ,forall n 2 Z + . Proof. For 1 i g ,wewrite i x [ z ]= X ` g X j =1 U ` i;j x z j V ` i;j x ; whereeach U i;j ;V i;j 2 C h x i .Set j;i y ; x = P ` U ` i;j T y V ` i;j x andnote X T I;I X = j;i X T I;I X g j;i =1 = X ` U ` i;j T X T I V ` i;j I X g j;i =1 ; forall X 2 M C g .Let 1 i g , vec [ n ] i X [ Z ]= vec [ n ] X ` g X j =1 U ` i;j X Z j V ` i;j X = X ` g X j =1 vec [ n ] U ` i;j X Z j V ` i;j X = g X j =1 X ` vec [ n ] Z j U ` i;j X T V ` i;j X ; 54

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wherethelastequalityisusingLemma6.1.Continuingon, vec [ n ] i X [ Z ]= g X j =1 X ` vec [ n ] Z j U ` i;j X T V ` i;j X = g X j =1 vec [ n ] Z j X ` U ` i;j T X T V ` i;j X = g X j =1 vec [ n ] Z j X ` U ` i;j T X T I V ` i;j I X = g X j =1 vec [ n ] Z j g X j =1 j;i X T I;I X = vec [ n ] Z X T I;I X i : Therefore, vec [ n ] X [ Z ]= vec [ n ] Z X T I;I X and vec [ n ] a X [ Z ]= vec [ n ] a x + vec [ n ] Z X T I;I X -1 forall n and X;Z 2 M n C g .Thus,itemiisproved.Foritemii,noteI )]TJ/F25 11.9552 Tf 1.02 0 0 1 210.669 396.052 Tm [(y ; x2 M gC h y x i,henceI )]TJ/F25 11.9552 Tf 1.02 0 0 1 391.9 396.052 Tm [(y ; x2 M gC < y x >.SinceI )]TJ/F25 11.9552 Tf 1.02 0 0 1 123.888 372.144 Tm [(;0=I,isinvertible,Proposition3.8in[19]impliesI )]TJ/F25 11.9552 Tf 1.02 0 0 1 429.223 372.144 Tm [(isinvertibleand I )]TJ/F25 11.9552 Tf 11.955 0 Td [( y ; x )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 2 M g C < y x > .Foritemiiiwenoten I m ;I n 0m=nm ;0nm2 dom nmI )]TJ/F25 11.9552 Tf 1.02 0 0 1 458.134 324.328 Tm [()]TJ/F28 7.9701 Tf 6.586 0 Td [(1 i;jsinceI nm )]TJ/F25 11.9552 Tf 11.956 0 Td [( nm ; 0 nm = I nm isinvertible. Itemivissimplyaconsequenceofiii; dom n 2 I )]TJ/F25 11.9552 Tf 11.955 0 Td [( )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 i;j dom I )]TJ/F25 11.9552 Tf 11.955 0 Td [( T; n )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 i;j : Finally,substitute n infor X and a n infor Z inEquation6-1toget vec [ n ] a n = vec [ n ] a n [ a n ] = vec [ n ] a n + vec [ n ] a n T; n : 55

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Hencebyrearranging, vec [ n ] a n I gn 2 )]TJ/F25 11.9552 Tf 11.955 0 Td [( T; n = vec [ n ] a n : Multiplyingbothsidesontherightby I gn 2 )]TJ/F25 11.9552 Tf 11.955 0 Td [( T; n )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 yields vec [ n ] a n = vec [ n ] a n I gn 2 )]TJ/F25 11.9552 Tf 11.955 0 Td [( T; n )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 ; asdesired. Ifaisahyporationalseriesthena[n]=a j M n C gisarationalfunctioningn 2commutingindeterminatesbyProposition6.1v.Infact,Proposition6.1showsthatasmallamountofcommutativityisallthepreventsahyporationalfrombeingrational. Denition6.3.Supposeax[z]=ax+x[z]ishyporealization.Let2 M gC h y x ibethematrixconstructedinProposition6.1.Wedenetobethehypomatrixrepresentation of a .Thatis, vec [ n ] a X [ Z ]= vec [ n ] a X + vec [ n ] Z I )]TJ/F25 11.9552 Tf 11.956 0 Td [( X T I;I X forall X;Z 2 M n C g and n 2 Z + . Let a 2 C hyp t x y behyporational.Dene dom n a = [ 2 A dom n I )]TJ/F25 11.9552 Tf 11.955 0 Td [( )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 ; where A isthecollectionofallhypomatrixrepresentationsof a .SupposeX 2 dom na.Letax[z]=ax+x[z]beahyporealizationsuchthataistherstcomponentofthesolutionofaandX 2 dom nI )]TJ/F25 11.9552 Tf 1.01 0 0 1 377.043 199.993 Tm [()]TJ/F28 7.9701 Tf 6.587 0 Td [(1,whereistheassociatedhypomatrixrepresentationof a .Wedene a X = vec [ n ] )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 vec [ n ] a X I )]TJ/F25 11.9552 Tf 11.955 0 Td [( X T I;I X )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 1 : Thus,wecanevaluate a atany X 2 dom n a . 56

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Example6.5. Let p x 1 ;x 2 ;x 3 = x 1 ;x 2 )]TJ/F30 11.9552 Tf 11.955 0 Td [(x 2 1 ;x 3 + x 1 )]TJ/F30 11.9552 Tf 9.298 0 Td [(x 2 + x 1 x 2 )]TJ/F30 11.9552 Tf 11.955 0 Td [(x 2 2 ,hence p x = x 0 B B B B B B @ 1 )]TJ/F30 11.9552 Tf 9.299 0 Td [(x 1 )]TJ/F30 11.9552 Tf 9.298 0 Td [(x 2 + x 1 x 2 01 )]TJ/F30 11.9552 Tf 9.299 0 Td [(x 2 001 1 C C C C C C A = x 0 B B B B B B @ 1 x 1 x 2 01 x 2 001 1 C C C C C C A )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 and p x [ z ]= x 1 ;x 2 + x 1 z 1 ;x 3 + x 1 z 2 + x 2 z 2 .Note p isahyporealizationandsetting y ; x = 0 B B B B B B @ 0 y 1 10 00 y 1 + y 2 1 000 1 C C C C C C A ; isthehypomatrixrepresentationof p .Observe isnilpotent,and I )]TJ/F25 11.9552 Tf 11.955 0 Td [( y ; x )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 = 0 B B B B B B @ 1 y 1 1 y 1 y 1 + y 2 1 01 y 1 + y 2 1 001 1 C C C C C C A ; and vec [ n ] n I )]TJ/F25 11.9552 Tf 11.955 0 Td [( T; n )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 = vec [ n ] q n : Thus q x = x 1 ;x 2 + x 2 1 ;x 3 + x 1 + x 2 x 2 + x 1 + x 2 x 2 1 . Example6.6.WeonceagainrevisitExample3.7.Thatis,px=x 1 ;x 2 )]TJ/F30 11.9552 Tf 12.93 0 Td [(x 1 x 2 x 1,qx=x 1 ; P 1 n =0 x n 1 x 2 x n 1andtheauxiliaryinverseofpis p x[z]=x 1 ;x 2+x 1 z 2 z 1.Sinceq 1x=x 1,^ p x[z]= x 1 ;x 2 + x 1 z 2 x 1 isahyporealizationwith^ p x[qx]=qx.Thehypomatrixrepresentationof ^ p is I )]TJ/F25 11.9552 Tf 11.955 0 Td [( y ; x = 0 B B @ 1 10 01 1 )]TJ/F30 11.9552 Tf 11.955 0 Td [(y 1 x 1 1 C C A ; withinverse, I )]TJ/F25 11.9552 Tf 11.955 0 Td [( y ; x )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 = 0 B B @ 1 10 0 1 )]TJ/F30 11.9552 Tf 11.955 0 Td [(y 1 x 1 )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 1 C C A : 57

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RecallthattheinverseofJ p,theJacobianmatrixofp,isapolynomialmatrix,howeverpisnotinjective.Inthiscase,thehypomatrixrepresentationwitnessesthenon-injectivityofp since I )]TJ/F25 11.9552 Tf 11.955 0 Td [( )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 isnotapolynomialmatrix.RecallseeLemma5.2thatifr 2 C < x >thenrt x2 C < x >t.SinceC < x >isaskeweld,C < x >tistheclassicalringofquotientsofC < x >[t].Hencethereexist; 2 C < x > [ t ] suchthat r t x = x [ t ] x [ t ] )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 andwecanextend deg t to C < x > t . Lemma6.2.Ifa 2C t x ygisatupleofhyporationalseriesthenthereexists2 Zsuchthatforall n 2 Z + , max n deg t a k t n i;j :1 k g; 1 i;j n o : -2 Proof.Letax[z]=ax+x[z]beahyporealizationofaandletbethehypomatrixrepresentationofa.Webeginbynotingthataisafreepolynomial,hencefor1 i;j nand1 k g,deg ta kt ni;j deg ta kt x:Set=max k f deg t a k t x g.Sincevec[n]preservestheentriesofmatrices, max 1 ` gn 2 n deg t vec [ n ] a t n ` o =max 1 k g 1 i;j n n deg t a k t n i;j o max 1 k g f deg t a k t x g = :NextwerecallI )]TJ/F25 11.9552 Tf 1.02 0 0 1 171.697 286.979 Tm [()]TJ/F28 7.9701 Tf 6.587 0 Td [(1 2 M gC < y [ x >andI )]TJ/F25 11.9552 Tf 1.02 0 0 1 342.737 286.979 Tm [(t T; n)]TJ/F28 7.9701 Tf 6.587 0 Td [(1existsforalln 2 Z +.Fornotationaleaseweset=I )]TJ/F25 11.9552 Tf 0.98 0 0 1 228.403 263.071 Tm [()]TJ/F28 7.9701 Tf 6.587 0 Td [(1.Eachi;jt y ;t x2 C < y x >t,thusbyProposition5.1weknowthereexistsd i;j 2 Z +suchthatforalln 2 Z +,deg ti;jt T; n d i;j.IfwesetD =max i;j f d i;j g ,thenforall n 2 Z + , max 1 k;` gn 2 deg t t T; n k;` D:Proposition6.1saysforeachn 2 Z +,vec[n]at nistheproductoftherowvectorvec[n]at nandthematrixt T; n.Thus,inlightofthedegreeboundsfoundabove58

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andLemma5.1,wehave max 1 k gn 2 n deg t vec [ n ] a t n k o + D:Finally,set=+Dandobserveoncemorethatsincevec[n]preservestheentriesofmatricesitmustpreservethedegreesoftheentries.Therefore,forall n 2 Z + , max n deg t a k t n i;j :1 k g; 1 i;j n o : Theorem6.7.Supposeaishyporational.Ifdom na=M nCgforalln,thenaisafreepolynomial. Proof.Thehyporationalityofaimpliesa j M n C gisamatrixofcommutativerationalfunctions.Infact,vec[n] a j M n C g vec[n])]TJ/F28 7.9701 Tf 6.587 0 Td [(1:C gn 2 ! C n 2isann 2-tupleofrationalfunctionsingn 2commutingindeterminates.Sincedom na=M nCg,eachvec[n] a j M n C g vec[n])]TJ/F28 7.9701 Tf 6.587 0 Td [(1kisarationalfunctionwithdomainofC gn 2,henceeachisapolynomial.Thus,eacha ni;jisapolynomialandinparticular,a nisapolynomial.However,Lemma6.2tellsusthatthereissome2 Z +suchthatforalln 2 Z +,max i;j;k f deg ta kt ni;jg.Thatis,aisafreeanalyticfunctionsuchthata nisapolynomialforeachn,andthedegreeofthepolynomialsisbounded.Therefore,Lemma5.3implies a isafreepolynomial. Corollary6.1.Afreepolynomialp:MCg ! MCgisbijectiveifandonlyifh ; x7! Dp x [ h ] ; x hasapolynomialinverse. Proof.Supposep:MCg ! MCgisafreepolynomialandFisitsscion.RecallFx ; y=Dpy[x]; y.IfFhasafreepolynomialinversethenFisbijective.HenceProposition4.1implies p isbijective.Supposepisbijective.AnapplicationofProposition4.1showsFisbijective.LetGbetheinverseofF.Theorem3.5impliesJ F,theJacobianmatrixofFisinvertibleasapolynomialmatrix.Hence,Gx ; y[z ; y]=x ; yJ )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 Fz ; y,theauxiliaryinverseofF,must59

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beapolynomialandbyLemma4.2,Gisahyporealization.Thus,Gisthesolutionofthehyporealization G x ; y [ z ; y ] and G ishyporational.SinceFisbijectiveandGisitsinverse,Lemma3.4saysG j M n C gagreeswithafreepolynomial,foreachn 2 Z +.Inparticular,dom nG=M nCgforeachn 2 Z +.Thus,Theorem6.7implies G isafreepolynomial. TheoremA-theFreeJacobianconjecture-tellsusthatafreepolynomialpisinjectiveifandonlyifDpY[X]isnonsingularforallX 2 MCg.Corollary6.1strengthensthiscondition. BijectivityCriteriaProposition4.1tellsusthatafreepolynomialisinjectiveifandonlyifitsscionisinjective.Thus,whentestingthebijectivityofafreepolynomial,itsucestoonlytestforthebijectivityofitsscion.Themainresultofthissection,Theorem6.9,combinesCorollary6.1withPascoe'sFreeJacobianconjecturetogetamoredirectanalogtotheclassicalJacobianconjecture.Letf 2C h y [ x ig.Wesayfisx-linearifj w j x=1forallmonomialswappearinginf.Inotherwords, f isasumofmonomialsthatcontainexactlyone x -term. Lemma6.3.Supposef 2C h y [ x ig.Iffisx-linear,thenthereexistsamatrixofbipartitepolynomials, J2 M g C h z y i ,suchthat vec [ n ] f Y [ X ]= vec [ n ] X J Y T I n ;I n Y ; forall X;Y 2 M n C g and n 2 Z + . Proof.WeomitthedetailsoftheconstructionofJsinceitisalmostexactlythesameastheconstructionofthehypomatrixrepresentationfoundinProposition6.1. Denition6.4.Supposep:MCg ! MCgisafreepolynomialwithderivative,Dpy[x]2C h y [ x ig.SinceDpy[x]isx-linear,Lemma6.3impliesthereexistsamatrix,60

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J hyp p 2 M g C h z y i ,suchthat vec [ n ] Dp Y [ X ]= vec [ n ] X J hyp p Y T I n ;I n Y ;forallX;Y 2 M nCgandn 2 Z +.WedeneJ hyp ptobethehypo-Jacobianmatrixofp.Thehypo-Jacobianmatrixisunique. Remark 6.8 . Hypo-Jacobianmatricessatisfythechainrule.Namely, J hyp z ; y = J hyp z ; y J hyp T z ; y ; forall ; 2 C h x i + g .AnyendomorphismofthefreeassociativealgebraC h x ihasaJacobianmatrixsee[6]and[27]thatexactlycorrespondswiththehypo-Jacobianmatrixfoundinthischapter.TheJacobianmatrixofanendomorphismisamatrixoverC h z i opp C h x i,whereC h z i oppistheoppositeringofC h z itheorderofmultiplicationisreversed.Theconstructionofthehypo-JacobianmatrixsendstermsoftheformUyx i VytoU Tz Vy.SincewecanviewthemapU 7! U Tasthecanonicalanti-isomorphismfromC h z i! C h z i opp,weseethatthehypo-JacobianmatrixofapolynomialmappingandtheJacobianmatrixofanendomorphismof C h x i areindeedthesame. Theorem6.9.ThefreepolynomialmappingpisinjectiveifandonlyifJ hyp phasamultiplicativeinversewhoseentriesarebipartitepolynomials. Proof.Supposepisinjective,qisitsinverseandF=Dpy[x]; yisthescionofp.LettingGbetheinverseofF,Lemma4.2showsGx ; y=Dqpy[x]; y.Corollary6.1impliesGisafreepolynomial,henceDqpy[x]isafreepolynomial.SinceDqpy[x]isx-linear,Lemma6.3impliesthereexistsamatrix J2 M g C h z y i ,suchthat vec [ n ] Dq p Y [ X ]= vec [ n ] X J Y T I n ;I n Y : 61

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Thechainruletellsus Dq p y [ x ]= Dq p y [ Dp y [ x ]]= x ,thus vec [ n ] X = vec [ n ] Dp Y [ X ] J Y T I n ;I n Y = vec [ n ] X J hyp p J Y T I n ;I n Y : Next, Dp q p y [ x ]= Dp y [ Dq p y [ x ]]= x ,hence vec [ n ] X = vec [ n ] Dq p Y [ X ] J hyp p Y T I n ;I n Y = vec [ n ] X J J hyp p Y T I n ;I n Y : Thus, vec [ n ] X = vec [ n ] X J hyp p J Y T I n ;I n Y = vec [ n ] X J J hyp p Y T I n ;I n Y ; forall X;Y 2 M n C g ,and n 2 Z + .Inotherwords, J hyp p )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 = J2 M g C h z y i . Converselysuppose J hyp p )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 2 M g C h z y i .Let ^ G bethefreepolynomialdenedby ^ G Y [ X ]= vec [ n ] )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 vec [ n ] X J hyp p )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 Y T I n ;I n Y ; forall X;Y 2 M n C g and n 2 Z + .Observe vec [ n ] X = vec [ n ] X J hyp p J hyp p )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 Y T I n ;I n Y = vec [ n ] Dp Y [ X ] J hyp p )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 Y T I n ;I n Y = vec [ n ] ^ G Y [ Dp Y [ X ]] : Hence, X = ^ G Y [ Dp Y [ X ]] and x = ^ G y [ Dp y [ x ]] .Ontheotherhand, vec [ n ] X = vec [ n ] X J hyp p )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 J hyp p Y T I n ;I n Y = vec [ n ] ^ G Y [ X ] J hyp p Y T I n ;I n Y = vec [ n ] Dp Y [ ^ G Y [ X ]] : 62

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Hence,X=DpY[^ GY[X]]andconsequently,x=Dpy[^ Gy[x]].BysettingG x ; y = ^ G y [ x ] ; y ,weget G F x ; y = G Dp y [ x ] ; y = ^ G y [ Dp y [ x ]] ; y = x ; y ; and F G x ; y = F ^ G y [ x ] ; y = Dp y [ ^ G y [ x ]] ; y = x ; y : Thus, G istheinverseof F .Therefore,byCorollary6.1, p isaninjectivefreepolynomial. BeforewenallymoveontotheproofofTheorem6.11weconnectthecompositionofpolynomialmappingstothecompositionofendomorphismsof C h x i . Denition6.5.Supposep 2C h x igisafreepolynomialmappingandlet:C h x i! C h x ibeanalgebrahomomorphism.Wesayisinducedby pifx i=p ix,for1 i g.Similarly,wesay p is inducedby if p x = x 1 ;:::; x g . Lemma6.4.Suppose;:C h x i! C h x iarealgebrahomomorphisms.Ifp;qaretheinducedpolynomialmappingsof and ,respectively,then q p x = x 1 ;:::; x g : Proof. Thisisveriedrathereasilyfromdenitions.Thedetailsarelefttothereader. IfisanendomorphismofC h x ithentheJacobianmatrixofisag gmatrixoverC h z i opp C h x i.Morespecically,ifpisthepolynomialmappinginducedby,thentheJacobianmatrixofisfoundbyapplyingthenaturalanti-isomorphismM gC h z i C h x i! M g C h z i opp C h x i to J hyp p . Theorem6.10.Supposep:MCg ! MCgisafreepolynomialmapping.Thefollowingareequivalent; i p isinjective; ii p isbijective; iii Dp Y isanonsingularmapforall Y 2 M n C g andall n 2 Z + ; 63

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iv J hyp p isinvertible; v p )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 existsandisafreepolynomial. Proof. i , ii , iiiisTheoremA.i , ivisTheorem6.9.v iisclear.Toshowivv,weassumeJ hyp pisinvertible.LetbetheendomorphismofC h x iinducedbypsox i=p ifor1 i g.SinceJ hyp pisinvertible,itfollowsthatthatJacobianmatrixofisinvertible.Thus,Proposition3.1in[6]impliesisanepicendomorphismandTheorem12.7in[27]impliesisanautomorphismofC h x i.So )]TJ/F28 7.9701 Tf 6.587 0 Td [(1existsandisanautomorphismitself. Let q = )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 x 1 ;:::; )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 x g bepolynomialmappinginducedby )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 .ByLemma6.4, x 1 ;:::;x g = )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 x 1 ;:::; )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 x g = q p x and x 1 ;:::;x g = )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 x 1 ;:::; )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 x g = p q x : Thus, p and q areinversemappings.Therefore, p isinjective. Theorem6.11FreeGrothendieckTheorem.Ifp:MCg ! MCgisaninjectivefreepolynomialmapping,thenthereisafreepolynomialmappingqsuchthatp qx=x=q px;thatis, p hasafreepolynomialinverse. Proof. Thisisexactlyi vinTheorem6.10. 64

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CHAPTER7 COMPUTINGINVERSESSupposep:MCg ! MCgisafreepolynomial,FisitsscionandqandGaretheinverseswhentheyexistofpandF,respectively.ByeitherCorollary6.1orTheorem6.9weknowthatpisinjectiveifandonlyifGisfreepolynomial.Recallthatifphasafreepolynomialinverseq,thenLemma4.2tellsusthatdegq degG degpdegq.Thus,anupperboundon deg G givesusanupperboundonthepossibledegreeof q . Denition7.1. Let V t =0 andforanynonzerorationalfunction r 2 C < x > dene V t r t x =min n max f deg t x [ t ] ; deg t x [ t ] g : r t x = x [ t ] x [ t ] )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 o : If M 2 M m n C < x > thendene V t M t x =max f V t M t x i;j :1 i m; 1 j n g : Notethatif r 2 C h x i then V t r t x =deg r x .Remark7.1.Itisstraightforwardtoseethatj deg t r t x j V trt x,foranynonzerorationalfunction.Byappealingtoevaluationsongenericmatriceswegetthat V t r t x s t x V t r t x + V t s t x and V t r t x + s t x V t r t x + V t s t x : Hence,if M 2 M ` m C < x > and N 2 M m n C < x > then V t M t x N t x max i;k m X j =1 V t M t x i;j N t x j;k max i;k m X j =1 V t M t x i;j + V t N t x j;k m V t M t x + m V t N t x : 65

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Lemma7.1.Letf=1andforanyintegern 1,denefn+1=n+13 fn.IfM;M )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 2 M n C < x > then j deg t M )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 t x j 3 n f n V t M t x . Proof. Weclaim V t M )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 t x 3 n f n V t M t x anduseinductiontoprovetheclaim.Ifn=1thenM;M )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 2 C < x >anddeg tMt x)]TJ/F28 7.9701 Tf 6.587 0 Td [(1=)]TJ/F25 11.9552 Tf 11.291 0 Td [(deg tMt x,henceV t M t x )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 = V t M t x 3 V t M t x . Nowsupposethestatementholdsfor n andconsider M;M )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 2 M n +1 C < x > .Write M = 0 B B @ Ab cd 1 C C A where A 2 M n C < x > , b 2 M n 1 C < x > , c 2 M 1 n C < x > ,and d 2 C < x > .IfS 2 GL n +1CthenMSisinvertible.Hence,wemayassumedisnonzeroandthus,d )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 exists.Observe V t A t x )]TJ/F30 11.9552 Tf 11.955 0 Td [(b t x d t x )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 c t x V t A + V t bd )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 c n +1 V t M t x : Hence V t A )]TJ/F30 11.9552 Tf 11.955 0 Td [(bd )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 c )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 3 n n +1 f n V t M : Set N t x = A t x )]TJ/F30 11.9552 Tf 11.955 0 Td [(b t x d t x )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 c t x )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 andobserve V t d )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 + cNbd )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 V t M + V t + cNbd )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 =2 V t M + V t cNb 2 V t M + n X i =1 V t c i + n X j;k =1 V t N j;k + V t b k 2 n +1 V t M + n 2 V t N : 66

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Applyingtheinductionhypothesis, 2 n +1 V t M + n 2 V t N n +1 V t M + n 2 n n +1 f n V t M = n +1 n 3 +3 n n 2 f n +2 n +2 V t M 3 n +1 n 3 + n 2 +2 n +2 f n V t M 3 n +1 n +1 3 f n V t M =3 n +1 f n +1 V t M :SincetheinverseofMisdeterminedfromtheSchurcomplement,wehaveproventhatV t M )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 t x 3 n f n V t M t x . Finally,since j deg t r t x j V t r t x foranynonzerorational r ,wehave deg t M )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 t x 3 n f n V t M t x ; asdesired. Lemma7.2. If M;M )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 2 M n C h x i then deg M )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 3 n f n deg M . Proof.SinceMandM )]TJ/F28 7.9701 Tf 6.587 0 Td [(1arematricesofpolynomials,V tMt x=degMt xandV t M t x )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 =deg M t x )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 .Thus,byLemma7.1, deg M t x )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 = deg t M t x )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 3 n f n V t M =3 n f n deg M : ThedegreeboundinLemma7.2isfarfromoptimal.However,toimprovethedegreeboundinasignicantmannerwouldrequireanaltogetherdierentproof;theinductionhypothesiscannotbeappliedtoA )]TJ/F30 11.9552 Tf 12.691 0 Td [(bd )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 c)]TJ/F28 7.9701 Tf 6.586 0 Td [(1sinceitisnotnecessarilytheinverseofapolynomialmatrix.SupposeB:N N ! NisafunctionsuchthatwheneverM 2 M nC h x iandM )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 2 M nC h x i,wehavedegM )]TJ/F28 7.9701 Tf 6.586 0 Td [(1Bn; degM.WecallsuchafunctionaPMID bound forPolynomialMatrixInverseDegree. 67

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Theorem7.2.SupposeBisaPMIDbound.Letpbeabijectivefreepolynomialandletqbeitsinverse.If q isafreepolynomialthen deg q B g; deg p )]TJ/F25 11.9552 Tf 11.955 0 Td [(1+1 . Proof.WebeginbynotingthatdegDpy[x]=degp.SinceJ hyp p,thehypo-Jacobianofp,isconstructedfrom Dp y [ x ] ,wegetthat deg J hyp p =deg p )]TJ/F25 11.9552 Tf 11.956 0 Td [(1 .ByTheorem6.10weknowJ hyp p)]TJ/F28 7.9701 Tf 6.587 0 Td [(1isapolynomialmatrixsincepisinjective.Hence,degJ hyp p)]TJ/F28 7.9701 Tf 6.587 0 Td [(1Bg; degp)]TJ/F25 11.9552 Tf 0.98 0 0 1 511.657 606.436 Tm [(1.Infact,forall n 2 Z + and X;Y 2 M n C g , vec [ n ] Dq p Y [ X ]= vec [ n ] X J hyp p )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 Y T I n ;I n Y :Thus,degDqpy[x]=degJ hyp p)]TJ/F28 7.9701 Tf 6.586 0 Td [(1+1Bg; degp)]TJ/F25 11.9552 Tf 1.02 0 0 1 394.876 510.802 Tm [(1+1.Lemma4.2saysG=Dqpy[x]; y,whereGistheinverseofF,thescionofp.Infact,degq degG.Therefore deg q deg G =deg Dq p y [ x ] B g; deg p )]TJ/F25 11.9552 Tf 11.955 0 Td [(1+1 ; asdesired. RecallfromDenition3.3, p x[z]=x J pzand p k +1x[z]= p kx[ p x[z]].Foreach k ,wewrite p k x [ z ]= X w 2h x [ z i k w w;setd k q=d z p k=inf n j w j : j w j z > 0 and k w 6 =0 oandwrite p kx[z]=q kx+r kx[z],where q k x = X w 2h x i j w j < d k q k w w and r k x [ z ]= X w 2h x [ z i j w j d k q k w w:FromLemma3.3weknowd k z >kandq k1 k =1isasequenceofpolynomialsconvergingtoqsuchthatdegq k% degq.Moreover,Lemma4.1showsthatqisapolynomialifandonlyifthereexistsan N suchthat deg p deg q N < d N z . Lemma7.3.SupposeBisaPMIDbound.Letp 2C h x i +gandletqbethecompositionalinverseofp.SetB=Bg; degp)]TJ/F25 11.9552 Tf 1.02 0 0 1 261.344 108.244 Tm [(1+1.ThenqisafreepolynomialifandonlyifJ )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 p 2 M g C h x i and deg q k B forall k . 68

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Proof.SupposeJ )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 p 2 M gC h x ianddegq k Bforallk.Inparticular,degq B deg p B,hence deg q B deg p deg p B deg p < d B deg p q and q isapolynomialbyLemma4.1.Converselysupposeqisapolynomial.ItfollowsthatqXexistsforallX 2 MCg,andconsequentlypqX=X=qpXforallX 2 MCg.Thus,pisbijectiveandTheorem3.5impliesJ )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 p 2 M gC h x i.Next,Theorem7.2impliesdegqBg; degp)]TJ/F25 11.9552 Tf 0.998 0 0 1 519.431 612.414 Tm [(1+1= B andsince deg q k deg q ,wehaveexactly deg q k B forall k . Lemma7.3hintsatasimplealgorithmfordeterminingwhetherq,theinverseofagivenpolynomial p 2 C h x i + g ,isapolynomial.Westillset B = B g; deg p )]TJ/F25 11.9552 Tf 11.956 0 Td [(1+1 . Corollary7.1. Either q B = q or q isnotapolynomialand p isnotinjective. Proof.Letq=P w w w.Recalldegq kisanincreasingsequenceandqandq kagreeonmonomialsoflengthlessthand k q.Ifq B 6=qthenthereexistsak>Bandw 2h x iwithj w j > d B q >Bsuchthat k w 6=0.Inparticulardegq k>B,thusbyLemma7.3,qisnotapolynomialand p isnotinjective. Forany p ,thealgorithmicapproachtocomputing q isasfollows. )]TJ0 g 0 G/F20 11.9552 Tf 26.361 0 Td [(If: J p = 2 M g C h x i then q isnotapolynomial; )]TJ0 g 0 G/F20 11.9552 Tf 26.361 0 Td [(Else:compute B = B g; deg p )]TJ/F25 11.9552 Tf 11.955 0 Td [(1+1 ; )]TJ0 g 0 G/F20 11.9552 Tf 26.361 0 Td [(Set k =1 and p 0 x [ z ]= z ; )]TJ0 g 0 G/F20 11.9552 Tf 26.361 0 Td [(Loop: Compute p k x [ z ]= p x [ p k )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 x [ z ]] ; If: deg q k >B or k>B then q isnotapolynomial; ElseIf: deg q k deg p < d k q then q = q ; Else:Increase k byone.InatmostBloops,theabovealgorithmwouldeithertellusthatqisnotapolynomialorwouldreturnapolynomialinverse q . 69

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APPENDIX ACOLLECTIONOFPROOFSInthisAppendixwesupplyproofsforafewkeyresultsthatwereeitherlefttothereaderorsourcedfromotherpapersbutwhoseproofsarestraightforwardenoughtonotrequiremucheorttoestablishthem.FirstupisaproofofLemma5.2.RecallthatinthesettingofLemma5.2,w=w 1 ;:::;w h isacollectionoffreelynoncommutingindeterminatesand w = w 1 ;:::;w h . Lemma5.2. Suppose r 2 C < w > .If t isacentralindeterminate,then r t w 2 C < w > t . Proof.Webeginbytakinganyr 2 C < w >,r 6=0,suchthatrt w2 C < w >t.ByRemark5.1weknowthereexists; 2 C < w >[t]suchthatrt w=w[t]w[t])]TJ/F28 7.9701 Tf 6.586 0 Td [(1.Thus, r t w )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 = w [ t ] w [ t ] )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 2 C < w > t .Wenowproceedbyinductionontheinversionheightofrepresentatives.Ifr=0thenrt u=02 C < w >tsowemayassumer 6=0.Ifarepresentativeofrhasaninversionheightof 0 then r isapolynomial,hence r t u 2 C h w i [ t ] C < w > t .Now,supposeifanyrationalfunction^ rhasarepresentativewithinversionheightn,then^ rt u2 C < w >t.Supposeisarepresentativeofrwithinversionheightn+1.Wecanwrite asasumofproductsofrationalexpressions; = X i m i Y j =1 i;j i;j )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 i;j ;whereeach i;j ; i;jand i;jarerationalexpressionswithinversionheightatmostn.Hence,each i;j ; i;jand i;jarerepresentativesofsomencrationalfunctions,a i;j ;b i;jandc i;j,andinfact,a i;jt u;b i;jt u;c i;jt u2 C < w >tbytheinductionhypothesis.Thus, b i;j t u )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 2 C < w > t and r t u = X i m i Y j =1 a i;j t u b i;j t u )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 c i;j t u 2 C < w > t : NextwegiveanexplicitproofofLemma5.3. 70

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Lemma5.3.Suppose:MCg ! MCgisafreeanalyticfunctionwithapowerseriesthatconvergesforeach X 2 M C g and n isapolynomialforeach n .Dene T = n deg t j t n k;` : n 2 Z + ; 1 j g; 1 k;` n o : If T isbounded,then isafreepolynomial. Proof. For 1 j g wewritethepowerseriesof j , j x = 1 X i =0 i j x ; whereeach i j x isahomogeneousfreepolynomialofdegree i .Hence j t n = deg n X i =0 i j n t i :Thereexists 2 Zsuchthat> max T,sinceTisbounded.Fori>wemusthave i j nk;`=0forall1 k;` n,bythedenitionofT.Hence i j n=0foralln 2 Z +,implying i j=0.Thus,hasnononzerotermsinitspowerseriesofdegreehigherthan,allowingustoconcludethat is,infact,afreepolynomial. Nextweprovethatacommutingrationalmapwithnopolesisapolynomial.Inordertodothiswemustrstintroducesomenotation.Supposeu=f u 1 ;:::;u m gisacollectionofcommutingindeterminatesandu=u 1 ;:::;u m.LetC[u]=C[u 1 ;:::;u m]denotethesetofallnonzeropolynomialsinC[u]=C[u 1 ;:::;u m].WedenetheoperationsofadditionandmultiplicationonC[u] C[u] inthefollowingway: ; ; := ; ; ; + ; := + ; : Moreover,if 6 =0 ,then ; )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 := ; . 71

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WedenetherelationonC[u] C[u]bystating;;ifandonlyif=.ItisstraightforwardtoverifythatisinfactanequivalencerelationonC[u] C[u].Wedene C u = C u 1 ;:::;u m := C [ u ] C [ u ] = .Wewillalmostalwayswriteacommutingrationalfunctionintermsofitsrepresentatives.Thatis,r= )]TJ/F28 7.9701 Tf 6.587 0 Td [(1meansristheequivalenceclassof;.Moreover,wewillsometimesexplicitlytalkaboutanelementofC[u] C[u]andwillwrite )]TJ/F28 7.9701 Tf 6.587 0 Td [(1inlieuof;.Contextwillallowustodistinguishwhetherwearediscussingelementsof C u or C [ u ] C [ u ] . If )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 2 C [ u ] C [ u ] thenwedene dom )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 ,thedomainof )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 tobe dom )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 = f c 2 C m : c 6 =0 g : Next,forany r 2 C u wedene dom r ,thedomainof r tobe dom r = n c 2 C m : 9 )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 2 r suchthat c 2 dom )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 o :Recallrisanequivalenceclass,so )]TJ/F28 7.9701 Tf 6.586 0 Td [(1 2 rmeans )]TJ/F28 7.9701 Tf 6.587 0 Td [(1isarepresentativeofr.Finally,ifc 2 domrthenrc,theevaluationofratcisgivenbyrc=cc)]TJ/F28 7.9701 Tf 6.587 0 Td [(1,where )]TJ/F28 7.9701 Tf 6.587 0 Td [(1isanyrepresentativeof r suchthat c 2 dom )]TJ/F28 7.9701 Tf 6.587 0 Td [(1 . LemmaA.4.Suppose; 2 C[u 1 ;:::;u m]arenonconstantpolynomials.Ifvanisheswhenever vanishes,then gcd ; isnotaconstant. Proof. Since C [ u 1 ;:::;u m ] isaUniqueFactorizationDomainwecanwrite = ap e 1 1 :::p e k k ; = bp f 1 1 :::p f k k ;wherea;b 2 C,thep i'sarenon-associateprimeelementsandatleastoneofe iandf jisnonzero.Becausevanisheswhenevervanishes,Hilbert'sNullstellensatzimpliesthereexistsn 2 Nand 2 C[u 1 ;:::;u m]suchthat n=.Choose1 i ksuchthatf i >0and72

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notethat j nimpliesp f i i j n,andinparticular,p i j n.Hencep i j sincep iisprime.Thus,p i dividesboth and ,therefore p i j gcd ; and gcd ; cannotbeaconstant. LemmaA.5.Supposer 2 Cu 1 ;:::;u m.Ifdom r=C mthenrisapolynomial.Thatis,acommutativerationalfunctionwithnopolesisapolynomial. Proof.Now,supposer 2 Cu 1 ;:::;u mhasadomainofC m.Weshowthatif )]TJ/F28 7.9701 Tf 6.586 0 Td [(1isarepresentativeofrthenvanisheswhenevervanishes.Accordingly,supposec 2 C msuchthatc=0.Sincedom r=C mthereexistsarepresentativeofr, )]TJ/F28 7.9701 Tf 6.586 0 Td [(1,suchthatc6=0.Hence, c c = c c =0 ,implying c =0 .Thus, vanisheswhenever vanishes.Let=gcd;,with=^ and=^ .Notethat^ ^ )]TJ/F28 7.9701 Tf 6.587 0 Td [(1isarepresentativeofrsince ^ =^ .Hence^ vanisheswhenever^ vanishes.Thefactthatgcd^ ; ^ =1andthecontrapositiveofLemmaA.4implyc6=0forallc 2 C m,i.e.isaconstant.Thus^ ^ )]TJ/F28 7.9701 Tf 6.586 0 Td [(1isapolynomialandtherefore r isapolynomial. 73

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BIOGRAPHICALSKETCHMericAugatisa30-year-oldmanwhohasspentmostofhisadultlifeearningadvanceddegreesandwritingparagraphsonrulednotebooksinhischildhoodbedroom.Heisselsh,unprincipled,hypocritical,gluttonous,manipulative,anddeluded|anunsavorycharacterover-all.Hedisdainsmodernity,particularlypopculture.Thedisdainhasbecomehisobsession:hegoestomoviesinordertomocktheirperversityandexpresshisoutragewiththecontemporaryworld'slackof"theologyandgeometry".AugatprefersthescholasticphilosophyoftheMiddleAges,andtheEarlyMedievalphilosopherBoethiusinparticular.Theworkingsofhispyloricvalveplayanimportantroleinhislife,reactingstronglytoincidentsinafashionthathelikenstoCassandraintermsofpropheticsignicance. 77