Algebraic approximation of analytic sets and mappi

arXiv:0712.3261v1[math.CV]19Dec2007AlgebraiapproximationofanalytisetsandmappingsMarinBilski1AbstratLet{Xν}beasequeneofanalytisetsonvergingtosomeanalytisetXinthesenseofholomorphihains.WeintrodueaonditionwhihimpliesthateveryirreduibleomponentofXisthelimitofasequeneofirreduibleomponentsofthesetsfrom{Xν}.Nextweapplytheonditiontoapproximateaholomorphisolutiony=f(x)ofasystemQ(x,y)=0ofNashequationsbyNashsolutions.Presentedmethodsallowtoonstrutanalgorithmofapproximationoftheholomorphisolutions.KeywordsAnalytimapping,analytiset,Nashset,approximationMSC(2000):Primary32C25,32C07;Seondary65H101IntrodutionLetKdenotetheeldofomplexorrealnumbers.Thefollowingapproximationtheoremisknowntobetrue:everyK-analytimappingf:Ω→KksuhthatQ(x,f(x))=0forx∈Ω,whereQisaK-Nashmapping(Ωdesribedbelow),anbeuniformlyapproximatedbyaK-NashmappingF:Ω→KksuhthatQ(x,F(x))=0forx∈Ω.IntheomplexasethetheoremwasprovedbyL.Lempert(see[16℄,The-orem3.2)foreveryholomorphiallyonvexompatsubsetΩofananealgebraivarietyandintherealaseitwasprovedbyM.Coste,J.RuizandM.Shiota(see[12℄,Theorem1.1)foreveryompatNashmanifoldΩ.Theapproximationtheoremturnedouttobeaverystrongtoolwithmanyimportantappliations(see[12℄,[16℄).Theproofsofthetheorempresentedin[12℄,[16℄relyonthesolutiontotheM.Artin’sonjeture:adeepresultfromommutativealgebraforwhihthereaderisreferredto[1℄,[17℄,[18℄,[19℄,[20℄.Suhanapproahenabledtoreahthegoalinanelegantandrelativelyshortway.Ontheotherhand,itseemstobeverydiulttoapplytheproofsinordertondNashapproximationsforonreteanalytimappingsheneitisnaturaltoaskwhetherthetheoremanbeobtaineddiretly.Thelatterquestionisstronglymotivatedbythefatthatapproximatinganalytiobjetsbyalgebraiounterpartsisoneofentraltehniquesusedinnumerialomputations.Fromthispointofviewit1M.Bilski:InstituteofMathematis,JagiellonianUniversity,Reymonta4,30-059Krakw,Poland.e-mail:Marin.Bilskiim.uj.edu.plThisisarevisedversionofthepapersubmittedforpubliationtoajournal(reeiptaknow-ledgedon30May2007).Manusriptrevisedon:20June2007,01Deember2007.ResearhpartiallysupportedbythePolishMinistryofSieneandHigherEduation.1isimportanttodeveloptheoryofapproximationthatouldserveasabaseforndingnumerialalgorithms.InSetion3.2ofthepresentpaperwegive,usingonlysomebasimethodsofanalytigeometry,aproofofasemi-globalversionofthetheoremintheomplexase(seeTheorem3.8).TheproofallowstoonstrutanalgorithmofapproximationofthemappingfwhihisdesribedinSubsetion3.2.4.ThefollowingloalversionisanimmediateonsequeneofTheorem3.8.Theorem1.1LetUbeanopensubsetofCnandletf:U→Ckbeaholo-morphimappingthatsatisesasystemofequationsQ(x,f(x))=0forx∈U.HereQisaNashmappingfromaneighborhoodˆUinCn×CkofthegraphoffintosomeCq.Thenforeveryx0∈UthereareanopenneighborhoodU0⊂Uandasequene{fν:U0→Ck}ofNashmappingsonverginguniformlytof|U0suhthatQ(x,fν(x))=0foreveryx∈U0andν∈N.IntheloalsituationtheproblemofapproximationofthesolutionsofalgebraioranalytiequationswasinvestigatedbyM.Artinin[2℄,[3℄,[4℄andTheorem1.1anbederivedfromhisresults.OurinterestinTheorem1.1anditsgeneralizationsispartiallymotivatedbyappliationsinthetheoryofanalytisets.Inpartiular,papers[6℄[10℄ontainavarietyofresultsonapproximationofomplexanalytisetsbyomplexNashsetswhoseproofsanbedividedintotwostages:(i)preparation,whereonlydiretgeometrimethodsappear,(ii)swithingTheorem1.1.Thusthetehniquesofthepresentartileallowtoobtainmanyoftheseresultsinapurelygeometriway.Asanexampleletusmentionthefollowingmaintheoremof[9℄.LetXbeananalytisubsetofpuredimensionnofanopensetU⊂CmandletEbeaNashsubsetofUsuhthatE⊂X.Thenforeverya∈EthereisanopenneighborhoodUaofainUandasequene{Xν}ofomplexNashsubsetsofUaofpuredimensionnonvergingtoX∩Uainthesenseofholomorphihainssuhthatthefollowingholdforeveryν∈N:E∩Ua⊂Xνandμx(Xν)=μx(X)forx∈(E∩Ua)\Fν,whereFνisanowheredenseanalytisubsetofE∩Ua.Hereμx(X)denotesthemultipliityofXatx(see[11℄,[13℄forthepropertiesandgeneralizationsofthisnotion).IntheproofofTheorem3.8weapplyTheorem3.1fromSetion3.1whih,beingofindependentinterest,istherstmainresultofthispaper.TheaimofSetion3.1istodevelopamethodofontrollingthebehaviorofirreduibleomponentsofanalytisetsfromasequene{Xν}onverginginthesenseofholomorphihainstosomeanalytisetX.Morepreisely,weformulateon-ditionswhihguaranteethatthenumbersoftheirreduibleomponentsofXandofXνareequalforalmostallνwhihintheonsideredontextimpliesthateveryirreduibleomponentofXisthelimitofasequeneofirreduibleomponentsofthesetsfrom{Xν}.Combining(theglobalversionof)Theorem1.1withTheorem3.1oneob-tainsanewmethodofalgebraiapproximationofanalytisetsextendingtheapproahof[6℄.Namely,letXbeananalytisubsetofU×Ckofpuredimen-sionnwithproperprojetionontotheRungedomainU⊂Cn.Itiswellknown2([24℄)thatXisasubsetofanotherpurelyn-dimensionalanalytisetX′givenbyX′={(x,z1,...,zk)∈U×Ck:p1(x,z1)=...=pk(x,zk)=0},wherepi∈O(U)[zi]isunitarywithnon-zerodisriminant,fori=1,...,k.Afterreplaingtheoeientsofpi,foreveryi,bytheirNashapproximationsonUweobtaintheset˜X′approximatingX′.Clearly,thisdoesnotmeanthatsomeomponentsof˜X′automatiallyapproximateX.Yet,byTheorem3.1thereisasystemofpolynomialequationssatisedbytheoeientsofpi,i=1,.