Topology of definable Hausdorff limits

Let $A\sub \R^{n+r}$ be a set definable in an o-minimal expansion $\S$ of the real field, $A' \sub \R^r$ be its projection, and assume that the non-empty fibers $A_a \sub \R^n$ are compact for all $a \in A'$ and uniformly bounded, {\em i.e.} all fibers are contained in a ball of fixed radius $B(0,R).$ If $L$ is the Hausdorff limit of a sequence of fibers $A_{a_i},$ we give an upper-bound for the Betti numbers $b_k(L)$ in terms of definable sets explicitly constructed from a fiber $A_a.$ In particular, this allows to establish effective complexity bounds in the semialgebraic case and in the Pfaffian case. In the Pfaffian setting, Gabrielov introduced the {\em relative closure} to construct the o-minimal structure \S_\pfaff generated by Pfaffian functions in a way that is adapted to complexity problems. Our results can be used to estimate the Betti numbers of a relative closure $(X,Y)_0$ in the special case where $Y$ is empty.Comment: Latex, 23 pages, no figures. v2: Many changes in the exposition and notations in an attempt to be clearer, references adde

Quantitative study of semi-Pfaffian sets

We study the topological complexity of sets defined using Khovanskii's Pfaffian functions, in terms of an appropriate notion of format for those sets. We consider semi- and sub-Pfaffian sets, but more generally any definable set in the o-minimal structure generated by the Pfaffian functions, using the construction of that structure via Gabrielov's notion of limit sets. All the results revolve around giving effective upper-bounds on the Betti numbers (for the singular homology) of those sets. Keywords: Pfaffian functions, fewnomials, o-minimal structures, tame topology, spectral sequences, Morse theory.Comment: Author's PhD thesis. Approx. 130 pages, no figure

On (2,3)-agreeable Box Societies

The notion of $(k,m)$-agreeable society was introduced by Deborah Berg et al.: a family of convex subsets of $\R^d$ is called $(k,m)$-agreeable if any subfamily of size $m$ contains at least one non-empty $k$-fold intersection. In that paper, the $(k,m)$-agreeability of a convex family was shown to imply the existence of a subfamily of size $\beta n$ with non-empty intersection, where $n$ is the size of the original family and $\beta\in[0,1]$ is an explicit constant depending only on $k,m$ and $d$. The quantity $\beta(k,m,d)$ is called the minimal \emph{agreement proportion} for a $(k,m)$-agreeable family in $\R^d$. If we only assume that the sets are convex, simple examples show that $\beta=0$ for $(k,m)$-agreeable families in $\R^d$ where $k<d$. In this paper, we introduce new techniques to find positive lower bounds when restricting our attention to families of $d$-boxes, i.e. cuboids with sides parallel to the coordinates hyperplanes. We derive explicit formulas for the first non-trivial case: the case of $(2,3)$-agreeable families of $d$-boxes with $d\geq 2$.Comment: 15 pages, 10 figure

Topological complexity of the relative closure of a semi-Pfaffian couple

Gabrielov introduced the notion of relative closure of a Pfaffian couple as an alternative construction of the o-minimal structure generated by Khovanskii's Pfaffian functions. In this paper, use the notion of format (or complexity) of a Pfaffian couple to derive explicit upper-bounds for the homology of its relative closure. Keywords: Pfaffian functions, fewnomials, o-minimal structures, Betti numbers.Comment: 12 pages, 1 figure. v3: Proofs and bounds have been slightly improve

Polynomial hierarchy, Betti numbers, and a real analogue of Toda’s theorem

Abstract. We study the relationship between the computational hardness of two well-studied problems in algorithmic semi-algebraic geometry – namely the problem of deciding sentences in the first order theory of reals with a constant number of quantifier alternations, and that of computing Betti numbers of semi-algebraic sets. We obtain a polynomial time reduction of the compact version of the first problem to the second. As a consequence we obtain an analogue of Toda’s theorem from discrete complexity theory for real Turing machines (in the sense of Blum, Shub and Smale)

On projections of semi-algebraic sets defined by few quadratic inequalities

Abstract. Let S ⊂ R k+m be a compact semi-algebraic set defined by P1 ≥ 0,..., Pℓ ≥ 0, where Pi ∈ R[X1,..., Xk, Y1,..., Ym], and deg(Pi) ≤ 2, 1 ≤ i ≤ ℓ. Let π denote the standard projection from R k+m onto R m. We prove that for any q&gt; 0, the sum of the first q Betti numbers of π(S) is bounded by (k + m) O(qℓ). We also present an algorithm for computing the the first q Betti numbers of π(S), whose complexity is (k + m) 2O(qℓ). For fixed q and ℓ, both the bounds are polynomial in k + m. 1

POLYNOMIAL HIERARCHY, BETTI NUMBERS AND A REAL ANALOGUE OF TODA’S THEOREM

Abstract. Toda [35] proved in 1989 that the (discrete) polynomial time hierarchy, PH, is contained in the class P #P, namely the class of languages that can be decided by a Turing machine in polynomial time given access to an oracle with the power to compute a function in the counting complexity class #P. This result which illustrates the power of counting is considered to be a seminal result in computational complexity theory. An analogous result in the complexity theory over the reals (in the sense of Blum-Shub-Smale real Turing machines [9]) has been missing so far. In this paper we formulate and prove a real analogue of Toda’s theorem. Unlike Toda’s proof in the discrete case, which relied on sophisticated combinatorial arguments, our proof is topological in nature. As a consequence of our techniques we are also able to relate the computational hardness of two extremely well-studied problems in algorithmic semi-algebraic geometry – namely the problem of deciding sentences in the first order theory of the reals with a constant number of quantifier alternations, and that of computing Betti numbers of semialgebraic sets. We obtain a polynomial time reduction of the compact version of the first problem to the second. This latter result might be of independent interest to researchers in algorithmic semi-algebraic geometry