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].