DIMACS Series in Discrete Mathematics and Theoretical Comput(2)

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

2ANDREIGABRIELOVANDTHIERRYZELL

Notations.IfXisasubsetofRn,we’lldenotebyX\Xitsfrontier.Unlessotherwisenoted,allsubsetsXareassumedtoberelativelycompact.

1.Semi-Pfa ansetsandrelativeclosure

Wewillrecallinthissectiontheusualde nitionsofPfa anfunctionsandsemi-Pfa ansetsasgivenbyKhovanski .Then,wewillde netherelativeclosureofasemi-Pfa ancouplethatappearsin[6].

Rnbeanopendomain.Thefollowing1.1.Pfa anfunctions.Let

de nitionsareduetoKhovanski [12](seealso[10,11]).

Definition1.Asequence(f1,...,f )offunctionsthatarede nedandana-

iscalledaPfa anchainifitsatis esonadi erentialsystemofthelyticin

form:

(1)dfi=n

j=1Pi,j(x,f1(x),...,fi(x))dxj,

whereeachPi,jisapolynomialinx,f1,...,fi,andthefollowingholds.

(P1)ThegraphΓioffiiscontainedinadomain ide nedbypolynomialin-

equalitiesin(x,f1(x),...,fi 1(x),t),andsuchthat Γi i.

(P2)Γiisaseparatingsubmanifoldin i,i.e. i\Γiisadisjointunionoftwo opensets +iand i.(See[12,p.38].ThisisalsocalledtheRolleleaf

conditionintheterminologyof[14,15].)

Ifα∈NisaboundonthedegreesofthepolynomialsPi,j,wesaythatthedegreeofthechainis(atmost)α.Theinteger iscalledthelengthofthechain.

APfa anfunctionq(x)withthechain(f1,...,f )isanyfunctionthatcanbewrittenas

q(x)=Q(x,f1(x),...,f (x)),

forsomepolynomialQ(x,y1,...,y ).IfthedegreeofQisβ,wesaythatthedegreeofthefunctionq(x)is(atmost)β.

Example2.Foranya=(a1,...,an)∈Rnthefunctionea:x→exp(a·x)isaPfa anfunctiononRn,since

dea(x)=n

j=1ajea(x)dxj.

Pfa anfunctionsformaratherlargeclassoffunctions.NoteinparticularthatelementaryfunctionscanbeseenasPfa anfunctionsonappropriatesubsetsoftheirdomainofde nition(seetheChapter1of[12]or[7]formoreprecisestatements).

tobeoftheformRn,Rn 1×R+,or(R+)n.Insections2and3,wewilltake

Inthatcase,wewillusethefollowingresultwhichisduetoKhovanski [12,p.79].Thereadercanalsorefertotherevisededition(inRussian)[13].

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