DIMACS Series in Discrete Mathematics and Theoretical Comput(5)

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...5Theorem12.ThestructureSiso-minimal.Moreover,ifXisade nablesetde nedbyaformulainvolvingthelimitsetsL1,...,LN,Xcanbepresentedasalimitsetwhoseformatisboundedbyane ectivefunctionoftheformatsofL1,...,LN.

2.Connectedcomponentsofalimitset

Inthissection,weestablishe ectiveboundsonthenumberofconnectedcom-ponentsoftherelativeclosureofasemi-Pfa ancouple(X,Y).Inwhatfollows,wewillalwaysassumethatYisnotempty.NotethatifYisempty,itfollowsfromDe nition8that( X)+mustbeemptytoo;then,XλiscompactandthenumberofconnectedcomponentsofX0isatmostthenumberofconnectedcomponentsofXλ.E ectiveboundsforthiscaseappearin[12,22].

Asannouncedintheintroduction,weassumeasin[6]thatXandYarerelativelycompact.Tokeeptheestimatessimple,we’llalsoassumethatthePfa anchain(f1,...,f )isde nedoverthewholeofRn×R+,sothattheboundgiveninTheorem3isapplicable.

2.1.Generalconsiderations.Weshowherehowtoreducetheproblemofcountingthenumberofconnectedcomponentsofalimitsettoaprobleminthesemi-Pfa ansetting.

LetΦbethe(squared)distancefunctiononRn×Rn:

Φ:Rn×Rn →R(7)(x,y)→|x y|2

Foranyλ>0,wecande nethedistancetoYλ,ΨλonXλby:

(8)Ψλ(x)=minΦ(x,y).y∈Yλ

ˇ:De nesimilarlyforx∈X

(9)Ψ(x)=minΦ(x,y).ˇy∈Y

Thefollowingresultappearsin[6,Theorem2.12].Theproofispresentedhereforthesakeofself-containment.

Theorem13.Let(X,Y)beasemi-Pfa ancouple.Then,thereexistsλ 1suchthatforeveryconnectedcomponentCof(X,Y)0,wecan ndaconnectedcomponentDλofthesetoflocalmaximaofΨλsuchthatDλisarbitrarilycloseto

C.

Proof.LetCbeaconnectedcomponentof(X,Y)0.Notethatbyde nitionˇSowemusthaveΦ(x,y)>0oftherelativeclosure,ifxisinC,itcannotbeinY.ˇ,andsinceYˇiscompact,wemusthaveΨ(x)>0.Also,anypointinforally∈Yˇ,henceΨ| C≡0.ˇbutnotin(X,Y)0.Sowemusthave C Y CmustbeinX,

ThismeansthattherestrictionofΨtoCtakesitsmaximuminsideofC.

Choosex0∈C,andletc=Ψ(x0)>0.Forasmallλ,thereisapointxλ∈Xλclosetox0suchthatcλ=Ψλ(xλ)isclosetoc,andisgreaterthanthemaximumofthevaluesofΨλoverpointsofXλcloseto C.Hencetheset{x∈Xλ|Ψλ(x)≥cλ}isnonempty,andtheconnectedcomponentAλofthissetthatcontainsxλisclosetoC.Thereexistsalocalmaximumx λ∈AλofΨλ.IfDλistheconnectedcomponentinthesetoflocalmaximaofΨλ,itiscontainedinZλandisclosetoC.

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