DIMACS Series in Discrete Mathematics and Theoretical Comput(4)

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

4ANDREIGABRIELOVANDTHIERRYZELL

1.4.Relativeclosure.¿Fromnowon,weconsidersemi-Pfa ansubsetsofR×R+witha xedPfa anchain(f1,...,f )inadomain.Wewrite(x1,...,xn)forthecoordinatesinRnandλforthelastcoordinate(whichwethinkofasapa-rameter.)IfXissuchasubset,weletXλ={x|(x,λ)∈X} RnandconsiderXasthefamilyofits bersXλ.ˇ=WeletX+=X∩{λ>0}andXn

Y)+=Y+;

( X)+ Y.

Then,theformatofthecouple(X,Y)isthecomponent-wisemaximumofthefor-matsofthefamiliesXandY.

Definition9.Let(X,Y)beasemi-Pfa ancouplein

ativeclosureof(X,Y)atλ=0by

ˇ\Yˇ ˇ.(6)(X,Y)0=X.Wede netherel-

Definition10.Let Rnbeanopendomain.Alimitsetin isasetoftheform(X1,Y1)0∪···∪(XK,YK)0,where(Xi,Yi)aresemi-Pfa ancouplesre-spectivelyde nedindomainsi Rn×R+,suchthatˇi= for1≤i≤K.Iftheformatsofthecouples(Xi,Yi)areboundedcomponent-wiseby(N,I,J,n, ,α,β),wesaythattheformatofthelimitsetis(K,N,I,J,n, ,α,β).

Example11.Any(notnecessarilyrestricted)semi-Pfa ansetXisalimitset.

oftheform(3).Proof.ItisenoughtoprovetheresultforabasicsetX

n={x∈R|gi(x)>0,1≤i≤r}.Letψ=ψ1···ψJandg=g1···gr.Assume

De nethesets W=(x,λ)∈X×Λ|g(x)>λ,|x|<λ 1; ×Λ| 1(x)=···= I(x)=0,ψ(x)=0,g(x)≥λ,|x|≤λ 1;Y1=(x,λ)∈ ×Λ| 1(x)=···= I(x)=0,g(x)=λ,|x|≤λ 1;Y2=(x,λ)∈ ×Λ| 1(x)=···= I(x)=0,g(x)≥λ,|x|=λ 1;Y3=(x,λ)∈

whereΛ=(0,1].IfY=Y1∪Y2∪Y3,itisclearthat(W,Y)satis estherequirementsofDe nition8.Thus(W,Y)isasemi-Pfa ancouple;itsrelativeclosureisX.

Foralln∈NweletSnbethecollectionoflimitsetsinRn,andS=∪n∈NSn.Thefollowingtheoremsumsuptheresultsin[6,Theorems2.9and5.1].

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