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)

On irreducible components of real exponential hypersurfaces

Fix any algebraic extension $\mathbb K$ of the field $\mathbb Q$ of rationals. In this article we study exponential sets $V\subset \mathbb R^n$. Such sets are described by the vanishing of so called exponential polynomials, i.e., polynomials with coefficients from $\mathbb K$, in $n$ variables, and in $n$ exponential functions. The complements of all exponential sets in $\mathbb R^n$ form a Noethrian topology on $\mathbb R^n$, which we will call Zariski topology. Let $P \in {\mathbb K}[X_1, \ldots ,X_n,U_1, \ldots ,U_n]$ be a polynomial such that $$V=\{ \mathbf{x}=(x_1, \ldots , x_n) \in \mathbb R^n| P(\mathbf{x}, e^{x_1}, \ldots ,e^{x_n})=0 \}.$$ The main result of this paper states that, under Schanuel's conjecture over the reals, an exponential set $V$ of codimension 1, for which the real algebraic set $\rm Zer(P)$ is irreducible over $\mathbb K$, either is irreducible (with respect to the Zariski topology) or every of its irreducible components of codimension 1 is a rational hyperplane through the origin. The family of all possible hyperplanes is determined by monomials of $P$. In the case of a single exponential (i.e., when $P$ is independent of $U_2, \ldots , U_n$) stronger statements are shown which are independent of Schanuel's conjecture.Comment: Some minor changes. Final version, to appear in Arnold Mathematical Journa