; TeX output 1998.09.24:1745 sʍl K`y cmr10MinimalUUtypGesinseparablyclosedelds(s獒 lZoGeUUChatzidakis kUniversitGeUUParis7,CNRS SCarolUUW*oGod [W*esleyanUUUniversity * s!", cmsy10xq"V cmbx100.pIn9troQduction.1Ina[1],examplesoftypGesofb> cmmi10U-rank1(i.e.,minimaltypGes)inthetheoriesofseparably closedeldswereconstructed,enroutetodisplayingcertaindimensionphenomena.W*econstructhereadditionalexampleswithU-rank1andofvqarioustranscendencedegreesoverIarbitraryseparablyclosedelds.OOurexamplesincludeoneswhichareminimalbutof1innitetranscendencedegree,i.e.,not1thin.OZOurinterestinbuildingnewexampleswasbpiquedafterseeingtheroleplayedbyminimaltypGesoverseparablyclosedeldsinHrushovski'sanalysisofabGelianvqarieties.fIn[2]Delonshows,inthesettingofseparablyclosedFeldsofnitedegreeofimpGerfection,IthatthereisaZariskigeometryonthesetofrealizationsofanyminimaltypGe.6Herproofgeneralizestothesettingofourexamplesaswell.QZThisarticleistheresultofseveralworkingsessionsbGetweentheauthorsatW*esleyanUniversityandParis7,1andwascompletedduringtheMoGdelTheoryofFieldsprogramat>MSRI>in1998.jEW*earegratefulforthehospitalityandsuppGortofallthreeinstitutions.W*ethankElisabGethBouscarenandFran9coiseDelonforreadinganearlierversionofthispapGer,UUprovidingusefulsuggestionsandcorrections.Bac9kground$onseparablyclosedelds.wLetjK!bGeaeldofcharacteristicp">0.ThenYK ^ 0er cmmi7pǫisasubeldofK ,ZisomorphictoKLuviatheF*robGeniusmapx1ɷ7!x^pR.1ThusYKisnaturally5aK ^pVn-vectorspace,;ofdimensioneitherinniteorp^eܫforsomenaturalnumbGere.The6degreeofimpGerfectionistakentobeinniteintherstcase,pandequaltoeotherwise. Fix9aprimenumbGer9p,ErandletL\=f+; ;;0;1g9bethelanguageofrings. sF*oreach elementeofN[f1g,wetakeS C Fe'KtobGetheL-theoryofseparablyclosedeldsoffcharacteristicpanddegreeofimpGerfectione.Ershov[4]showedthateachS C Feiscomplete,andthatallcompletionsofthetheoryofseparablyclosedeldsareofthisform.F*urthermore,Banycompletetheoryofseparablyclosedeldsisstable(WoGod[5]),BandissupGerstableUUonlywhenitisthetheoryofalgebraicallyclosedelds(i.e.,whene=0).̍ LetRKnbGeaeldofcharacteristicp.ʾW*esaythatBԘKnisp-indepGendentiftheset cofJallp-monomialsinBM۫,i.e.,monomialsJoftheformb:iٓR cmr7(1)l1;ib:i(n)]"n%withb1|s;:::;bnط2BZB%and0=Mi(1);:::;i(n)ph( 1,isBlinearlyindepGendentintheK ^pVn-vectorspaceK .FA0maximalp-indepGendentUUsubsetB0ofKqiscalledap-basis,andinthiscaseK~4=K ^pVn[BM۫]. F*oreachnxanenumerationmi;n ʫ( x)ofthep-monomialsx:i(1)l1;ix:i(n)]"nwith0/i(1);:::;i(n)p8 1,UUanddenethe(n8+1)-aryUUfunctionsi;n l:K ^naz8K~4!KqasUUfollows:Ifthen-tuple\qMBbisnotp-indepGendent,orifthe(no+1)-tuple(\qUbJ;a)isp-independent,theni;n ʫ(\qUbJ;a)=0.qOtherwise,UUthei;n(\qUbJ;a)satisfy`* ^$a=prO \ cmmi5nlO! cmsy7 1獍/͟u cmex10Xti=0i;n ʫ(\qUbJ;a)pRmi;n(\qUb):dNote_]thatthesefunctionsdepGendontheeldK ,andthattheabove_]propertiesdenethemEuniquely*.BThesefunctionsarealsodenableinthepureringlanguage.ConsidertheM@languageL =dIL'[fi;n jn2N;0i