# Algebraic functions

## Algebraicfunctions

Algebraicfunctionsrefertoaclassoffullyanalyticfunctions.Referstothemulti-valuedfunctiondeterminedbytheirreducibleequation:

,whereaj(z)(j=0,1,...,n)isthepolynomialofz.Fromthealgebraicequationofw,weknowthatmultiplevalues​​ofwaredeterminedforeachvalueofz,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.

## 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

(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.