Abstract. The notion of relative closure (X, Y)0 of a semi-Pfaffian couple (X, Y) was introduced by Gabrielov to give a description of the o-minimal structure generated by Pfaffian functions. In this paper, an effective bound is given for the number of con
6ANDREIGABRIELOVANDTHIERRYZELL
2.2.Aboundforthesmoothcase.WewillnowshowhowthenumberofconnectedcomponentsofthesetoflocalmaximaofΨλthatappearinTheorem13canbeestimatedwhenthesetsXλandYλaresmooth.
De neforallp,
(10)
where
(11)p={(x,y0,...,yp)∈Xλ×(Yλ)p+1|yi=yj,0≤i<j≤p}.Wλpp={(x,y0,...,yp)∈Wλ|Φ(x,y0)=···=Φ(x,yp)},Zλ
Lemma14.Assume(X,Y)isaPfa ancouplesuchthatXλandYλaresmoothforallλ>0.Foragivenλ>0,letx bealocalmaximumofΨλ(x).p ,...,yp)∈ZλsuchThen,thereexists0≤p≤dim(Xλ)andapointz =(x ,y0ppthatZλissmoothatz ,andz isacriticalpointofΦ(x,y0)onZλ.
∈YλProof.Sincex isalocalmaximumofΨλ(x),thereexistsapointy0 suchthatΦ(x,y0)=miny∈YλΦ(x,y)=Ψλ(x).Inparticular,dyΦ(x,y)=0at ).If(x ,y0)isacriticalpointofΦ(x,y)(thisisalwaysthecase(x,y)=(x ,y0 )=0atwhendim(Xλ)=0)thestatementholdsforp=0.OtherwisedxΦ(x,y0 )(ξ)>0.x=x .LetξbeatangentvectortoXatx suchthatdxΦ(x ,y0 Assumethatforally∈YλsuchthatΦ(x,y)=Ψλ(x),wehavedxΦ(x,y)(ξ)>
˙(0)=ξ.For0whenx=x .Letγ(t)beacurveonXλsuchthatγ(0)=x andγ
ally∈Yλ,thereexistsTysuchthatforall0<t<Ty,theinequalityΦ(γ(t),y)>Φ(x ,y)holds.BycompactnessofYλ,thismeanswecan ndsometsuchthatthatinequalityholdsforally∈Yλ.Hence,Ψλ(γ(t))>Ψλ(x ),whichcontradictsthehypothesisthatΨλhasalocalmaximumatx . ∈YλsuchSincex isalocalmaximumofΨλ(x),thereexistsapointy1 =thatdxΦ(x,y1)(ξ)≤0atx=xandΦ(x,y1)=Ψλ(x).Inparticular,y1 y0,dyΦ(x,y)=0aty=y1,anddxΦ(x,y1)=dxΦ(x,y0)atx=x.This 11 ,y1)∈Zλ,andZλissmoothat(x ,y0,y1).If(x ,y0,y1)isimpliesthat(x ,y01acriticalpointofΦ(x,y0)onZλ(thisisalwaysthecasewhendim(Xλ)=1) )anddxΦ(x,y1)arelinearlythestatementholdsforp=1.OtherwisedxΦ(x,y0 independentatx=x.Sincedim(Xλ)≥2,thereexistsatangentvectorξtoXλ atx suchthatdxΦ(x ,y0)(ξ)>0anddxΦ(x ,y0)(ξ)>0.Sincex isalocal ∈YλsuchthatdxΦ(x,y2)(ξ)≤0atmaximumofΨλ(x),thereexistsapointy2 22isx=xandΦ(x,y2)=Ψλ(x).Thisimpliesthat(x,y0,y1,y2)∈Zλ,andZλ 23smoothat(x,y0,y1,y2).TheaboveargumentscanberepeatednowforZλ,Zλ,etc.,toprovethestatementforallp≤dim(Xλ).
AssumenowthatXλandYλaree ectivelynon-singular,i.e.theyareofthefollowingform:
(12)Xλ={x∈Rn|p1(x,λ)=···=pn d(x,λ)=0};
Yλ={y∈Rn|q1(y,λ)=···=qn k(y,λ)=0};
where,forallλ>0,weassumethatdxp1∧···∧dxpn d=0onXλandthatdyq1∧···∧dyqn k=0onYλ.Inparticular,wehavedim(Xλ)=danddim(Yλ)=k.Remark.Notethatweassumethatnoinequalitiesappearin(12).WecanclearlymakethatassumptionforYλ,sincethatsethastobeclosedforallλ>0.ForXλ,p,thecriticalsetofweobservethefollowing:ifCisaconnectedcomponentofCλ