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We introduce two versions of proof systems dealing with systems of inequalities: Positivstellensatz refutations and Positivstellensatz calculus. For both systems we prove the lower bounds on degrees and lengths of derivations for the example due to Lazard, Mora and Philippon. These bounds are sharp, as well as they are for the Nullstellensatz refutations and for the polynomial calculus. The bounds demonstrate a gap between the Null-and Positivstellensatz refutations on one hand, and the polynomial calculus and Positivstellensatz calculus on the other.
Let X be a subset in [−1, 1] n 0 ⊂ R n 0 defined by a formulawhere Q i ∈ {∃, ∀}, Q i = Q i+1 , x i ∈ R n i , and Xν be either an open or a closed set in [−1, 1] n 0 +...+nν being a difference between a finite CW -complex and its subcomplex. We express an upper bound on each Betti number of X via a sum of Betti numbers of some sets defined by quantifier-free formulae involving Xν .In important particular cases of semialgebraic and semi-Pfaffian sets defined by quantifier-free formulae with polynomials and Pfaffian functions respectively, upper bounds on Betti numbers of Xν are well known. Our results allow to extend the bounds to sets defined with quantifiers, in particular to sub-Pfaffian sets.
We prove new upper bounds on homotopy and homology groups of o-minimal sets in terms of their approximations by compact o-minimal sets. In particular, we improve the known upper bounds on Betti numbers of semialgebraic sets defined by quantifier-free formulae, and obtain for the first time a singly exponential bound on Betti numbers of sub-Pfaffian sets.Definition 3.7. Let B be a simplex in S. Fix some m ≥ 0 and a sequence ε 0 , δ 0 , ε 1 , δ 1 , . . . , ε m , δ m . Define V B as the union of sets K B ′ (δ i , ε i ) over all simplices B ′ ∈ S B , simplices K of R such that B ⊂ K, and i = 0, . . . , m. Define V as the union of the sets V B over all simplices B of S.
Topological relations between V and SLemma 4.
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