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1、TRANSACTIONSOFTHEAMERICANMATHEMATICALSOCIETYVolume348,Number8,August1996ONMULTIPLICITIESINPOLYNOMIALSYSTEMSOLVINGM.G.MARINARI,H.M.MOLLER,ANDT.MORA¨Abstract.Thispaperdealswiththedescriptionofthesolutionsofzerodi-mensionalsystemsofpolynomialequations.Basedondi eren
2、tmodelsfordescribingsolutions,weconsidersuitablerepresentationsofamultipleroot,ormorepreciselysuitabledescriptionsoftheprimarycomponentofthesystemataroot.Weanalysethecomplexityof ndingtherepresentationsandofalgo-rithmswhichperformtransformationsbetweenthedi erent
3、representations.IntroductionWhensolvingasystemofpolynomialequations(whichinthispaperwillalwayshavea0-dimensionalsetofzeroesandwillbecalled-dimensionalsystem"),itisoftensatisfactorytoknowthezeroesofthesystem,insuchawaytobeabletoperformarithmeticaloperationsontheco
4、ordinatesofeachroot.Thereisofcourseastreamofresearchaboutmethodsforsolvingsystemsofequations(forasurveywereferto[L93]).Thereisalsoa"reflection"aboutthemeaning"ofsolvingasystem,takingplacewithinsomeresearchgroupsmoreinterestedine ectivemethodsforalgebraicgeometry.
5、ThephilosophyessentiallygoesbacktoKronecker:asystemissolvedifeachrootisrepresentedinsuchawayastoallowtheperformanceofanyarithmeticaloperationsoverthearithmeticalexpressionsofitscoordinates(theoperationsinclude,intherealcase,numericalinterpolation).Forinstance,int
6、heclassicalKroneckermethod,concerningtheunivariatecase,oneisgivenatowerofalgebraic eldextensionsofthe eldofrationalnumbers,each eldbeingapolynomialringoverthepreviousonemodulotheidealgeneratedbyasinglepolynomialandeachrootisrepresentedbyanelementinsuch elds.Thema
7、ine ortoftheactualresearchisdevotedtoe ectivetechniquesforrepresentingrootsofasystemandallowingecientarithmeticaloperationsovertheirexpressions.Inthiscontextonecouldhoweverbeinterestedalsointhemultiplicityofeachroot,notjustintheweakrithmetical"senseofsimple,doub
8、le,triple,etc.root,butinthestrongeralgebraic"senseofgivingasuitabledescriptionoftheprimarycomponentatarootoftheidealde ningthesolutionsetofthesystem.Theaimoft
