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Betti Numbers of Semialgebraic and Sub-Pfaffian Sets

Authors:	Gabrielov A; Vorobjov N; Zell T

Institution:	Department of Mathematics, Purdue University West Lafayette, IN 47907, USA, agabriel{at}math.purdue.edu Department of Computer Science, University of Bath Bath BA2 7AY, nnv{at}cs.bath.ac.uk Department of Mathematics, Purdue University West Lafayette, IN 47907, USA, tzell{at}math.purdue.edu

Abstract:	Let X be a subset in –1,1]^n₀ $sub$ R^n₀ defined by the formula X={x₀\|Q₁x₁Q₂x₂...Qx ((x₀,x₁,...x) $isin$ X)}, where Q_i $isin$ { $exist$ ${forall}$ }, Q_i $!=$ Q_i+1, x_i $isin$ –1, 1]^n_i, and X may be eitheran open or a closed set in –1,1]^n₀+...+n, being the differencebetween a finite CW-complex and its subcomplex. An upper boundon each Betti number of X is expressed via a sum of Betti numbersof some sets defined by quantifier-free formulae involving X. In important particular cases of semialgebraic and semi-Pfaffiansets defined by quantifier-free formulae with polynomials andPfaffian functions respectively, upper bounds on Betti numbersof X are well known. The results allow to extend the boundsto sets defined with quantifiers, in particular to sub-Pfaffiansets.

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