Algebraicfunctions
Algebraicfunctionsrefertoaclassoffullyanalyticfunctions.Referstothemulti-valuedfunctiondeterminedbytheirreducibleequation:
,whereaj(z)(j=0,1,...,n)isthepolynomialofz.Fromthealgebraicequationofw,weknowthatmultiplevaluesofwaredeterminedforeachvalueofz,sow=w(z)isamulti-valuedfunction.AnalgebraicfunctionisacompleteanalyticfunctionwithonlyafinitenumberofalgebraicfulcrumsandpolesontheextendedcomplexplaneC^;onthecontrary,acompleteanalyticfunctionwiththeabovecharacteristicsmustsatisfyanirreduciblealgebraicequationandremoveanon-zeroconstantfactorOutsidethisequationisunique.TheRiemannsurfacecorrespondingtothealgebraicfunctioniscompact,thatis,aclosedsurface.Thegenusofthissurfaceisdefinedasthegenusofthealgebraicfunction.TheintegraloftherationalfunctionR(z,w)ofzandwconnectedbyequation(1):
iscalledtheAbelianintegral,wherethevalueofw(z)Itisderivedfromtheanalysisanddevelopmentofthebranchselectedbythez0pointalongtheintegrationpath.Itisamulti-valuedfunction,anditsmulti-valueisnotonlyproducedbytheresidualofR(z,w),themulti-valueofw(z),butalsodependsonthetopologicalpropertiesofthecorrespondingRiemannsurfaceofw(z).Forthisintegral,peopleoftenlookforaseriesofstandardforms,sothatanyofthistypeofintegralcanbetransformedintooneofthestandardformsthroughappropriatevariabletransformations.
TheresearchontheAbelianintegralleadstotheproblemofsingularizationofalgebraicfunctions,andthesingularizationofalgebraicfunctionsleadstothedevelopmentofgeneralsingularizationtheory.Inthisregard,fromthesecondhalfofthe19thcenturytothefirsttenyearsofthe20thcentury,manyfamousmathematiciansintheworldsuchasRiemann(GFB),Klein(C.)F.),Poincaré,J.-H.),Schwarz(Schwarz,HA),Neumann(CG)andKebe(Koebe,P.)haveallmadeimportantcontributions.
Developmenthistory
The(multi-valued)analyticfunctionw=w(z)iscalledanalgebraicfunction.AlgebraicfunctiontheorybeganwiththestudyofellipticfunctionsbyGauss,Abel,andJacobiintheearly19thcentury.WiththeestablishmentofthebasisoffunctiontheorybyRiemannandWeilstrass,itformedacompletetheory.Historically,algebraicfunctiontheoryhasdevelopedinthreedifferentdirections.TheequationP(z,w)=0determinesthecurveinthetwo-dimensionalcomplexprojectivespacewithzandwasthecoordinates.Fromthisperspective,theresearchbeganwithRiemann,ClayBush,Goldinandothers,afterTheworkofBrill,M.Nott,Severi,SegreandothersoftheItalianSchoolhasbeenlinkedtomodernalgebraicgeometry.Inthisera,thenumberfunctionisregardedasarationalfunctiononthealgebraicfamily,soitisstudiedbythemethodofalgebraicgeometry.Theso-called"analyticalmethod"tostudyalgebraicfunctionsasfunctionsonRiemannsurfaces(asRiemannsurfacesandmeromorphicfunctionsoncomplexmanifolds)isthebasicideaofRiemann,AbelandWeilstrass,ItwasinheritedbyCFKleinandHilbert,andfurtherorganizedbyWeylintoaperfectandrigorousform.ThepurealgebraicmethodofstudyingalgebraicfunctionsthroughthealgebraicfunctiondomainbeganintheresearchofDedekinandWeberattheendofthe19thcentury.Withthedevelopmentofabstractalgebraintheearly20thcentury,thisdirectionhasachievedtheoriesincludinggeneralcoefficientdomainsandcomplexvariablealgebraicfunctiontheory.Manyresults.Peoplealsoparticularlyrecognizethesimilaritiesbetweenalgebraicfunctiontheoryandnumbertheory,sotheresearchofalgebraicfunctiontheoryalsopromotesthedevelopmentofnumbertheory.Theabovethreedifferentviewpointswereinitiallynotonlymanifestedinthedifferentmethodsandexpressionstheyadopted,butalsointheterminologytheyused.However,withthepassageoftime,peoplehavediscoveredthatwiththedevelopmentofalgebraicmethods,manyoftheresultsfirstobtainedwithfunctionaltheoryandgeometricmethods,ifthealgebraicanalogsofthesemethodsareused,theycanoftenbesuccessfullyappliedtomoregeneraldomains.Circumstance,sothesedifferenceshavebecomeirrelevant.
Application
Inthemiddleandlate20thcentury,withtherapiddevelopmentofcomputerscienceandtechnology,thefactorizationofmultivariatepolynomialsisconsideredtobetheoriginofthefieldofsymboliccomputing.Thefactorizationofmultivariatepolynomialsisoneofthebasiccontentsinalgebra,andalsooneoftheimportantcontentsofmathematicsresearch.Itisnotonlyoneofthemostdifficultproblemsinmathematics,butalsothemostbasicalgorithminsymboliccalculation.Inmoderncomputeralgebrasystems,thecalculationofpolynomialfactorizationinthealgebraicalgebraicfunctiondomainhasaveryimportantposition.Atpresent,theresearchonthefactorizationofpolynomialsinthealgebraicnumberfieldisrelativelycomplete.Itiseasytooperateintermsoftherealizationofthealgorithmandtheefficiencyofthealgorithm.Therefore,manyfactorizationalgorithmsonthealgebraicnumberfieldhavebeenproposedbythepredecessors.Allhavebeenwidelyused,suchasthealgorithmproposedbyBarryM.Tragerin1976.However,withthecontinuousdeepeningofmathematicalresearch,itisnotsoeasytofactorizethealgebraicfunctiondomain.Itnotonlyhasahugeamountofcalculation,butalsoThespecificoperationofthealgorithmisalsomorecomplicated.Therefore,exploringthefactorizationalgorithmofmultivariatepolynomialsinthealgebraicfunctiondomainnotonlyhastheoreticalsignificance,butalsohasveryimportantapplicationvalue.
Analyticfunction
Alsoknownasholomorphicfunctionorregularfunction,itisthemainresearchobjectofanalyticfunctiontheory.Forthesingle-valuedfunctionf(z)ofthecomplexvariablezdefinedintheareaDonthecomplexplane,ifitisinaneighborhoodofeachpointz0inD,youcanusezz0meansthatf(z)isparsedinD.Weierstrass(K.(TW))startsfromthepowerseriesandestablishestheseriestheoryofanalyticfunctions.IfateachpointzinD,thelimitis:
(calledthederivativeoffunctionf(z)atpointz)exists.Cauchy(A.-L.)saidthatf(z)isanalyticinD.Thesetwodefinitionsareequivalent.Thefunctionf(z)=u(x,y)+iv(x,y),anotherequivalentconditionofz=x+iyinDis:u=u(x,y),v=v(x,y)Thereisacontinuouspartialderivativeateachpointz=x+iyinD,anditsatisfiestheCauchy-Riemannequation(orCauchy-Riemanncondition):
ThisconditionissometimesreferredtoasCRconditionorD'Alembert-Eulercondition.Thefourthequivalentconditionfortheanalysisofthefunctionf(z)intheregionDisMoreira'stheorem.
Analyticfunctionreferstoafunctionthatcanbelocallyexpandedintoapowerseries,anditisthemainobjectofresearchonthetheoryofcomplexvariables.Theanalyticfunctionclassincludesmostofthefunctionsencounteredinmathematicsanditsapplicationsinnaturalscienceandtechnology.Thebasicoperationsofarithmetic,algebra,andanalysisofthistypeoffunctionareclosed,andtheanalyticfunctionisinthedomainofitsnaturalexistence.Representstheonlyfunction,therefore,thestudyofanalyticfunctionsisofspecialimportance.
Thesystematicstudyofanalyticfunctionsbeganinthe18thcentury.Eulerhasmademanycontributionsinthisregard.Lagrangefirsthopedtoestablishasystematicanalyticfunctiontheory.Hetriedtodevelopthistheorybyusingthetoolsofpowerseries,buthewasunsuccessful.
TheFrenchmathematicianCauchyisrecognizedasthefounderofanalyticfunctiontheorywithhisownwork.In1814,hedefinedtheregularfunctionastheexistenceandcontinuityofthederivative.Hecriticizedmanywrongresultsinthepastandcreatedanumberoflawstoensurethereliabilityoftheseriesoperation.In1825,heobtainedthefamousCauchyintegraltheorem,andthenestablishedtheCauchyintegralformula.Cauchyusedthesetoolstoobtaintheresultthattheregularfunctioncanbeexpressedasaconvergentpowerserieseverywhereinitsdomain,anditsinversepropositionisalsotrue.Soparsingandregularizationareequivalent.LaterRiemannmadeimportantdevelopmentstoCauchy'swork.In1900,theFrenchmathematicianGulsaimprovedthedefinitionofregularfunctions,onlyrequiringthefunctiontohavederivativeseverywhereinthedomainofdefinition.
Weilstrassstartedthestudyofanalyticfunctionswiththepowerseriesasthestartingpoint.Hedefinedaregularfunctionasafunctionthatcanbeexpandedintoapowerseries,createdtheanalyticaldevelopmenttheory,andusedanalyticdevelopmenttodefineacompleteanalyticalfunction.Cauchy'smethodislimitedtotheso-calledsingle-valuedbranchofthecompleteanalyticfunction,anditmustbeunifiedwithWeylstrass'theorythroughanalyticaldevelopment.