DIMACS Series in Discrete Mathematics and Theoretical Comput(9)

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...9can ndacouple(x ,y )∈Xλ×(Yi)λcloseto(x ,y )suchthatΦ(x ,y )isalocalmaximumofthedistance(measuredbyΦ)fromXλto(Yi)λ.

Sinceforsmallenoughε0,...,εN,thesetsXλand(Yi)λaree ectivelynon-singularhypersurfaces,thenumberoflocalmaximaofthedistanceofXλto(Yi)λcanbeboundedby(16),forappropriatevaluesoftheparameters.Theestimate(19)follows.

3.Applicationtofewnomials

Inthissection,wewillapplyourpreviousresultstothecasewherethePfa anfunctionsweconsiderarefewnomials.

3.1.Fewnomialsandlowadditivecomplexity.Recallthatwecancon-sidertherestrictionofanypolynomialqtoanorthantasaPfa anfunctionwhosecomplexitydependsonlyonthenumberofnonzeromonomialsinq.FixK={m1,...,mr}∈Nnasetofexponents.

Definition18.ThepolynomialqisaK-fewnomialifitisoftheform:

q(x)=a0+a1xm1+···arxmr,wherea0,...,ar∈R.

Let =n+r,and(f1,...,f )bethefunctionsde nedby: 1x if1≤i≤n,i(20)fi(x)=xmi nifi>n.

Itiseasytoseethat(f1,...,f )isaPfa anchainoflength anddegreeα=2in

=(R+)n,sincewehave:thepositiveorthant

fi

S∩ .Notethatthisisindeedan

importantcase,sinceanexampleisgivenin[4]ofafewnomialsemialgebraicsetforwhichtheclosurecannotbedescribedindependentlyofthedegreesofthede ningpolynomials.

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