DIMACS Series in Discrete Mathematics and Theoretical Comput(3)

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...3Theorem3.Let(f1,...,f )beaPfa anchainon,whereisoneofRn,Rn 1×R+,or(R+)n.Ifq1,...,qnarePfa anfunctionsofrespectivedegreesβi,thenumberofsolutionsofthesystem

q1(x)=···=qn(x)=0,

forwhichtheJacobiandeterminantisnotzeroisboundedby

(2)2 ( 1)/2β1···βn[β1+···+βn n+min(n, )α+1] .

isnotRemark.Itispossibletogivee ectiveestimateswhenthedomain

oftheaboveform,butonehastomodifytheestimatestotakeintoaccountthecomplexityofthedomain.

1.2.Semi-Pfa ansets.Let(f1,...,f )beaPfa anchainofdegreeα,

Rn.de nedonadomain

Definition4.Abasicsemi-Pfa ansetisasetXoftheform:

(3)X={x∈| i(x)=0,ψj(x)>0,fori=1,..,I;j=1,..,J},

whereallthefunctionsabovearePfa anwiththechain(f1,...,f ).Ifallthesefunctionshavedegreeatmostβ,thesetXissaidtohaveformat(I,J,n, ,α,β).

Asemi-Pfa ansetisany niteunionofbasicsemi-Pfa ansets.Asemi-Pfa ansethasformat(N,I,J,n, ,α,β)ifitistheunionofatmostNbasicsemi-Pfa ansetshavingformatscomponent-wisenotexceeding(I,J,n, ,α,β).

Wesaythatasemi-Pfa ansetXisrestrictedifitisrelativelycompactin.Definition5.Abasicsemi-Pfa ansetiscalled(e ectively)non-singularifforallx∈X,thefunctions iappearingin(3)verify

d 1(x)∧···∧d I(x)=0.

1.3.E ectiveboundsontheBettinumbers.FollowingideasofOle nik-Petrovski [17],Milnor[16]andThom[20]foralgebraicvarieties,wecanusetheboundappearinginTheorem3toestimatethesumoftheBettinumbersofasemi-Pfa anset[12,22].

AssumeXisacompactsemi-Pfa ansetde nedonlybyequations,

(4)X={x∈Rn|p1(x)=···=pr(x)=0},

2andletp=p21+···+pr.Then,itcanbeshownthatthesumoftheBettinumbers

ofXisexactlyone-halfthesumoftheBettinumbersofthecompactcomponentsoftheset{p=ε},whereεisasmallpositiverealnumber.

Countingthenumberofcriticalpointsofagenericprojection,weobtainthefollowingbound.

Corollary6.AssumeX Rnisoftheform(4),wherep1,...,prareof

=RnwithalengthdegreeatmostβinaPfa anchain(f1,...,f )de nedin

anddegreeα.Then,thesumoftheBettinumbersofXisboundedby

(5)2 ( 1)/2β(α+2β 1)n 1[(2n 1)β+(n 1)(α 2)+min(n, )α] .

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