On the number of homotopy types of fibres of a definable map

In this paper we prove a single exponential upper bound on the number of possible homotopy types of the fibres of a Pfaffian map, in terms of the format of its graph. In particular we show that if a semi-algebraic set $S \subset {\R}^{m+n}$, where $\R$ is a real closed field, is defined by a Boolean formula with $s$ polynomials of degrees less than $d$, and $\pi: {\R}^{m+n} \to {\R}^n$ is the projection on a subspace, then the number of different homotopy types of fibres of $\pi$ does not exceed $s^{2(m+1)n}(2^m nd)^{O(nm)}$. As applications of our main results we prove single exponential bounds on the number of homotopy types of semi-algebraic sets defined by fewnomials, and by polynomials with bounded additive complexity. We also prove single exponential upper bounds on the radii of balls guaranteeing local contractibility for semi-algebraic sets defined by polynomials with integer coefficients.Comment: Improved combinatorial complexit

Approximation of definable sets by compact families, and upper bounds on homotopy and homology

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.Comment: 20 pages, 2 figure

Complexity of Computing the Local Dimension of a Semialgebraic Set

AbstractThe paper describes several algorithms related to a problem of computing the local dimension of a semialgebraic set. Let a semialgebraic set V be defined by a system of k inequalities of the formf≥ 0 with f∈R [ X1,⋯ ,Xn ], deg(f) <d , andx∈V . An algorithm is constructed for computing the dimension of the Zariski tangent space to V at x in time (kd)O(n). Let x belong to a stratum of codimension lxin V with respect to a smooth stratification ofV . Another algorithm computes the local dimension dimx(V) with the complexity (k(lx+ 1)d)O(lx2n). Ifl=maxx∈Vlx, and for every connected component the local dimension is the same at each point, then the algorithm computes the dimension of every connected component with complexity (k(l+ 1)d)O(l2n). If V is a real algebraic variety defined by a system of equations, then the complexity of the algorithm is less thankdO(l2n) , and the algorithm also finds the dimension of the tangent space to V at x in time kdO(n). Whenl is fixed, like in the case of a smooth V , the complexity bounds for computing the local dimension are (kd)O(n)andkdO(n) respectively. A third algorithm finds the singular locus ofV in time (kd)O(n2)

Bounds of some real (complex) solution of a finite system of polynomial equations with rational coefficients

We discuss two conjectures. (I) For each x_1,...,x_n \in R (C) there exist y_1,...,y_n \in R (C) such that \forall i \in {1,...,n} |y_i| \leq 2^{2^{n-2}} \forall i \in {1,...,n} (x_i=1 \Rightarrow y_i=1) \forall i,j,k \in {1,...,n} (x_i+x_j=x_k \Rightarrow y_i+y_j=y_k) \forall i,j,k \in {1,...,n} (x_i \cdot x_j=x_k \Rightarrow y_i \cdot y_j=y_k) (II) Let G be an additive subgroup of C. Then for each x_1,...,x_n \in G there exist y_1,...,y_n \in G \cap Q such that \forall i \in {1,...,n} |y_i| \leq 2^{n-1} \forall i \in {1,...,n} (x_i=1 \Rightarrow y_i=1) \forall i,j,k \in {1,...,n} (x_i+x_j=x_k \Rightarrow y_i+y_j=y_k)Comment: LaTeX2e, 28 pages, a shortened and revised version will appear in Mathematical Logic Quarterly 56 (2010), no.2, under the title ``Two conjectures on the arithmetic in R and C'

Complexity of deciding whether a tropical linear prevariety is a tropical variety.

We give an algorithm, with a singly exponential complexity, deciding whether a tropical linear prevariety is a tropical linear variety. The algorithm relies on a criterion to be a tropical linear variety in terms of a duality between the tropical orthogonalization A ⊥ and the double tropical orthogonalization A ⊥ ⊥ of a subset A of the vector space (R∪ { ∞}) n. We also give an example of a countable family of tropical hyperplanes such that their intersection is not a tropical prevariety. </p