DIMACS Series in Discrete Mathematics and Theoretical Comput(7)

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

NUMBEROFCONNECTEDCOMPONENTSOFTHERELATIVECLOSURE...7p,thefunctionΦisconstantonC.IfCcontainsalocalmaximumforΨ,itΦ|Zλλcannotmeet Xλbecause Xλ Yλ.Hence,wedonotneedtotakeintoaccounttheinequalitiesappearinginthede nitionofXλ.

Letusnowde neforallp,

(13)θp:(y0,...,yp)∈(Yλ)p+1→

0≤i<j≤p|yi yj|2.

parede nedforallpbythefollowingThen,forXλandYλasin(12),thesetsZλconditions. p1(x,λ)=···=pn d(x,λ)=0;

(14)q1(yi,λ)=···=qn k(yi,λ)=0,0≤i≤p; Φ(x,yi) Φ(x,yj)=0,0≤i<j≤p;

andtheinequality

(15)θp(y0,...,yp)>0.

Underthesehypotheses,weobtainthefollowingbound.

Theorem15.Let(X,Y)beasemi-Pfa ancouplesuchthatforallsmallλ>0,XλandYλaree ectivelynon-singularbasicsetsofdimensionrespectivelydandk.Iftheformatof(X,Y)is(1,I,J,n, ,α,β),thenumberofconnectedcomponentsof(X,Y)0isboundedby

(16)2d

p=02q (q 1)/2βp(α+2βp)nq 1[nq(2βp+α 2)+qmin(n, )α]q ,

whereq=p+2andβp=max{1+(n k)(α+β 1),1+(n d+p)(α+β 1)}.

Proof.AccordingtoTheorem13,wecanchoseλ>0suchthatforanyconnectedcomponentCof(X,Y)0,wecan ndaconnectedcomponentDλofthesetoflocalmaximaofΨλsuchthatDλisclosetoC.Weseethatforλsmallenough,twoconnectedcomponentsCandC of(X,Y)0cannotsharethesameˇforλsmallenough.Indeed,theconnectedcomponentDλ,sinceDλcannotmeetYˇisboundedfrombelowbythedistancefromDλtoYλ,–distancefromDλtoYˇButthelatterdistancewhichisatleastcλ,–minusthedistancebetweenYλandY.

goestozero,whereastheformergoestoapositiveconstantcwhenλgoestozero.

Oncethatλis xed,allweneedtodoisestimatethenumberofconnectedcomponentsofthesetoflocalmaximaofΨλ.AccordingtoLemma14,wecanpreducetoestimatingthenumberofconnectedcomponentsofthecriticalsetsCλ

pfor0≤λ≤d.oftherestrictionΦ|Zλ

Forthesakeofconcision,wewilldropλfromthenotationsinthisproof,p,pi(x)forpi(x,λ),etc...writingZpforZλ

Apointz=(x,y0,...,yp)∈ZpisinCpifandonlyifthefollowingconditionsaresatis ed: dyΦ(x,yj)=0,0≤j≤p;(17)rank(dxΦ(x,y0),...,dxΦ(x,yp))≤p.

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