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
8ANDREIGABRIELOVANDTHIERRYZELL
ForXλandYλasin(12),thoseconditionstranslateinto:
(18) rank{ yq1(yi),..., yqn k(yi), yΦ(x,yi)}≤n k,0≤i≤p;
rank{ xp1(x),..., xpn d(x), xΦ(x,y0),..., xΦ(x,y0)}≤n d+p.
Thoseconditionstranslateintoallthemaximalminorsofthecorrespondingmatri-cesvanishing.TheseminorsarePfa anfunctionsinthechainusedtode neXandY.Theirdegreesarerespectively1+(n k)(α+β 1)and1+(n d+p)(α+β 1).
ThenumberofconnectedcomponentsofCpisboundedbythenumberofconnectedcomponentsofthesetDpde nedbytheconditionsin(14)and(15),andthevanishingofthemaximalminorscorrespondingtotheconditionsin(18).
LetEpbethesetde nedbytheequations(14)and(18),sothatDp=Ep∩{θp>0}.Then,thenumberofconnectedcomponentsofDpisboundedbythenumberofconnectedcomponentsofEpplusthenumberofconnectedcomponentsofEp∩{θp=ε}forachoiceofε>0smallenough.
Hence,we’rereducedtotheproblemofestimatingthenumberofconnectedcomponentsoftwosemi-Pfa ansetsinRn(p+2)thatarede nedwithoutinequalitiesusingaPfa anchaininn(p+2)variablesofdegreeαandlength(p+2) .UsingtheboundsontheBettinumbersfromCorollary6,weobtain(16).
2.3.Boundsforthesingularcase.Let’sconsidernowthecasewhereXλandYλmaybesingular.Wecanusedeformationtechniquestoreducetothesmoothcase.First,thefollowinglemmashowswecanreducetothecasewhereXλisabasicset.
Lemma16.LetX1,X2andYbesemi-Pfa ansetssuchthat(X1,Y)and(X2,Y)arePfa anfamilies.Then,(X1∪X2,Y)0=(X1,Y)0∪(X2,Y)0.
Theprooffollowsfromthede nitionoftherelativeclosure.
Theorem17.Let(X,Y)beasemi-Pfa ancouple.AssumeXλandYλareunionsofbasicsetshavingaformatoftheform(I,J,n, ,α,β).IfthenumberofbasicsetsinXλisMandthenumberofbasicsetsinYλisN,thenthenumberofconnectedcomponentsof(X,Y)0isboundedby
(19)2MNn 1
p=02q (q 1)/2γp(α+2γp)nq 1[nq(2γp+α 2)+qmin(n, )α]q ,
whereq=p+2andγp=1+(p+1)(α+2β 1).
Proof.Again,wewanttoestimatethenumberoflocalmaximaofthefunctionΨλde nedin(8).
ByLemma16,wecanrestrictourselvestothecasewhereXisbasic.LetY=Y1∪···∪YN,whereallthesetsYiarebasic.Foreachbasicset,wetakethesumofsquaresoftheequationsde ningit:thecorrespondingpositivefunctions,whichwedenotebypandq1,...,qN,havedegree2βinthechain.Fixεi>0,for0≤i≤N,andλ>0,andletX={p(x,λ)=ε0}andforall1≤i≤N,letYi={qi(x,λ)=εi}.SinceYλiscompact,ifx isapointinXλsuchthatΨλhasalocalmaximumatx=x ,thereisapointy insome(Yi)λsuchthatΦ(x,y)=Ψλ(x).Then,we