Algebraic varieties with semialgebraic universal cover

We study projective varieties whose universal cover is biholomorphic to a semialgebraic open subset of a projective variety

Łojasiewicz exponent of overdetermined semialgebraic mappings

We prove that both local and global Łojasiewicz exponent of a continuous overdetermined semialgebraic mapping F : X → Rᵐ on a closed semialgebraic set X ⊂ Rⁿ (i.e. m > dimX) are equal to the Łojasiewicz exponent of the composition L ₒ F : X → Rᵏ for the generic linear mapping L : Rᵐ → Rᵏ, where k = dimX

Erdos-Szekeres-type statements: Ramsey function and decidability in dimension 1

A classical and widely used lemma of Erdos and Szekeres asserts that for every n there exists N such that every N-term sequence a of real numbers contains an n-term increasing subsequence or an n-term nondecreasing subsequence; quantitatively, the smallest N with this property equals (n-1)^2+1. In the setting of the present paper, we express this lemma by saying that the set of predicates Phi={x_1<x_2,x_1\ge x_2}$ is Erdos-Szekeres with Ramsey function ES_Phi(n)=(n-1)^2+1. In general, we consider an arbitrary finite set Phi={Phi_1,...,Phi_m} of semialgebraic predicates, meaning that each Phi_j=Phi_j(x_1,...,x_k) is a Boolean combination of polynomial equations and inequalities in some number k of real variables. We define Phi to be Erdos-Szekeres if for every n there exists N such that each N-term sequence a of real numbers has an n-term subsequence b such that at least one of the Phi_j holds everywhere on b, which means that Phi_j(b_{i_1},...,b_{i_k}) holds for every choice of indices i_1,i_2,...,i_k, 1<=i_1<i_2<... <i_k<= n. We write ES_Phi(n) for the smallest N with the above property. We prove two main results. First, the Ramsey functions in this setting are at most doubly exponential (and sometimes they are indeed doubly exponential): for every Phi that is Erd\H{o}s--Szekeres, there is a constant C such that ES_Phi(n) < exp(exp(Cn)). Second, there is an algorithm that, given Phi, decides whether it is Erdos-Szekeres; thus, one-dimensional Erdos-Szekeres-style theorems can in principle be proved automatically.Comment: minor fixes of the previous version. to appear in Duke Math.

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