We study forking, Lascar strong types, Keisler measures and definable groups, under an assumption of NIP (not the independence property), continuing aspects of the paper [16]. Among key results are (i) if $p=tp(b/A)$ does not fork over $A$ then the Lascar strong type of $b$ over $A$ coincides with the compact strong type of $b$ over $A$ and any global nonforking extension of $p$ is Borel definable over $bdd\left(A\right)$, (ii) analogous statements for Keisler measures and definable groups, including the fact that $G^{000}=G^{00}$ for $G$ definably amenable,...

The purpose of this paper is to carry over to the o-minimal settings some results about the Euler characteristic of algebraic and analytic sets. Consider a polynomially bounded o-minimal structure on the field ℝ of reals. A ($C^{\infty }$) smooth definable function φ: U → ℝ on an open set U in ℝⁿ determines two closed subsets W := u ∈ U: φ(u) ≤ 0, Z := u ∈ U: φ(u) = 0. We shall investigate the links of the sets W and Z at the points u ∈ U, which are well defined up to a definable homeomorphism. It is proven...

A local-global version of the implicit function theorem in o-minimal structures and a generalization of the theorem of Wilkie on covering open sets by open cells are proven.

Here we prove a Poincaré - Verdier duality theorem for the o-minimal sheaf cohomology with definably compact supports of definably normal, definably locally compact spaces in an arbitrary o-minimal structure.

We prove some properties similar to the theorem Ax-Kochen-Ershov, in some cases of pairs of algebraically maximal fields of residue characteristic p > 0. This properties hold in particular for pairs of Kaplansky fields of equal characteristic, formally p-adic fields and finitely ramified fields. From that we derive results about decidability of such extensions.

This paper investigates the geometry of the expansion ${}_Q$ of the real field ℝ by restricted quasianalytic functions. The main purpose is to establish quantifier elimination, description of definable functions by terms, the valuation property and preparation theorem (in the sense of Parusiński-Lion-Rolin). To this end, we study non-standard models of the universal diagram T of ${}_Q$ in the language ℒ augmented by the names of rational powers. Our approach makes no appeal to the Weierstrass preparation...

A coordinate cone in ${ℝ}^n$ is an intersection of some coordinate hyperplanes and open coordinate half-spaces. A semi-monotone set is an open bounded subset of ${ℝ}^n$, definable in an o-minimal structure over the reals, such that its intersection with any translation of any coordinate cone is connected. This notion can be viewed as a generalization of convexity. Semi-monotone sets have a number of interesting geometric and combinatorial properties. The main result of the paper is that every semi-monotone...

We study the action of G = SL(2,ℝ), viewed as a group definable in the structure M = (ℝ,+,×), on its type space $S_G\left(M\right)$. We identify a minimal closed G-flow I and an idempotent r ∈ I (with respect to the Ellis semigroup structure * on $S_G\left(M\right)$). We also show that the “Ellis group” (r*I,*) is nontrivial, in fact it is the group with two elements, yielding a negative answer to a question of Newelski.

A super real closed ring is a commutative ring equipped with the operation of all continuous functions ℝⁿ → ℝ. Examples are rings of continuous functions and super real fields attached to z-prime ideals in the sense of Dales and Woodin. We prove that super real closed rings which are fields are an elementary class of real closed fields which carry all o-minimal expansions of the real field in a natural way. The main part of the paper develops the commutative algebra of super real closed rings, by...

Let $ℰ$ be a disjoint decomposition of ${ℝ}^n$ and let $X$ be a vector field on ${ℝ}^n$, defined to be linear on each cell of the decomposition $ℰ$. Under some natural assumptions, we show how to associate a semiflow to $X$ and prove that such semiflow belongs to the o-minimal structure ${ℝ}_{\mathrm{an},exp}$. In particular, when $X$ is a continuous vector field and $\Gamma$ is an invariant subset of $X$, our result implies that if $\Gamma$ is non-spiralling then the Poincaré first return map associated $\Gamma$ is also in ${ℝ}_{\mathrm{an},exp}$.

We consider the expansion of the real field by the group of rational points of an elliptic curve over the rational numbers. We prove a completeness result, followed by a quantifier elimination result. Moreover we show that open sets definable in that structure are semialgebraic.

In answering questions of J. Maříková [Fund. Math. 209 (2010)] we prove a triangulation result that is of independent interest. In more detail, let R be an o-minimal field with a proper convex subring V, and let st: V → k be the corresponding standard part map. Under a mild assumption on (R,V) we show that a definable set X ⊆ Vⁿ admits a triangulation that induces a triangulation of its standard part st X ⊆ kⁿ.

We prove that the set of asymptotic critical values of a $C^1$ function definable in an o-minimal structure is finite, even if the structure is not polynomially bounded. As a consequence, the function is a locally trivial fibration over the complement of this set.

We give some examples of polynomially bounded o-minimal expansions of the ordered field of real numbers where the Weierstrass division theorem does not hold in the ring of germs, at the origin of ℝⁿ, of definable $C^{\infty }$ functions.

Given an o-minimal expansion ℳ of a real closed field R which is not polynomially bounded. Let ${}^{\infty }$ denote the definable indefinitely Peano differentiable functions. If we further assume that ℳ admits ${}^{\infty }$ cell decomposition, each definable closed subset A of Rⁿ is the zero-set of a ${}^{\infty }$ function f:Rⁿ → R. This implies ${}^{\infty }$ approximation of definable continuous functions and gluing of ${}^{\infty }$ functions defined on closed definable sets.