O-minimal Research Articles

Definable linear orders definably embed into lexicographic orders in o-minimal structures

We classify definable linear orders in o-minimal structures expanding groups. For example, let ( P , ≺ ) (P,\prec ) be a linear order definable in the real field. Then ( P , ≺ ) (P,\prec ) embeds definably in ( R n + 1 , > lex ) (\mathbb {R}^{n+1},>_{\text {lex}}) , where > lex >_{\text {lex}} is the lexicographic order and n n is the o-minimal dimension of P P . This improves a result of Onshuus and Steinhorn in the o-minimal group context.

Monotone functions and maps

In [S. Basu, A. Gabrielov, N. Vorobjov, Semi-monotone sets. arXiv:1004.5047v2 (2011)] we defined semi-monotone sets, as open bounded sets, definable in an o-minimal structure over the reals, and having connected intersections with all translated coordinate cones in R^n. In this paper we develop this theory further by defining monotone functions and maps, and studying their fundamental geometric properties. We prove several equivalent conditions for a bounded continuous definable function or map to be monotone. We show that the class of graphs of monotone maps is closed under intersections with affine coordinate subspaces and projections to coordinate subspaces. We prove that the graph of a monotone map is a topologically regular cell. These results generalize and expand the corresponding results obtained in Basu et al. for semi-monotone sets.

First order tameness of measures

We develop a general framework for measure theory and integration theory that is compatible with o-minimality. Therefore the following natural definitions are introduced. Given are an o-minimal structure M and a Borel measure μ on some Rn. We say that μ is M-compatible if there is an o-minimal expansion of M such that for every parameterized family of subsets of Rn that is definable in M the corresponding family of μ-measures is definable in this o-minimal expansion. We say that μ is M-tame if there is an o-minimal expansion of M such that for every parameterized family of functions on Rn that is definable in M the corresponding family of integrals with respect to μ is definable in this o-minimal expansion. We substantiate these definitions with existing and many new examples. We investigate the Lebesgue measure in their light. We prove definable versions of fundamental results such as the theorem of Radon–Nikodym, the Lebesgue decomposition theorem and the Riesz representation theorem. They allow us to describe explicitly compatible and tame measures and to classify them in dimension one.

Locally o-minimal structures

In this paper we study (strongly) locally o-minimal structures. We first give a characterization of the strong local o-minimality. We also investigate locally o-minimal expansions of (R, +, <).

Elementarily equivalent structures for topological languages over regions in Euclidean space

We prove that the class of rational polyhedra and the class of topologically regular regions definable in an o-minimal structure over the reals are each elementarily equivalent to the class of polyhedra for topological languages.

Connected components of definable groups and [formula omitted]-minimality I

We give examples of definable groups G (in a saturated model, sometimes o-minimal) such that G00≠G000, yielding also new examples of “non G-compact” theories. We also prove that for G definable in a (saturated) o-minimal structure, G has a “bounded orbit” (i.e. there is a type of G whose stabilizer has bounded index) if and only if G is definably amenable, giving a positive answer to a conjecture of Newelski and Petrykowski in this special case of groups definable in o-minimal structures.

Hausdorff measure on o-minimal structures

AbstractWe introduce the Hausdorff measure for definable sets in an o-minimal structure, and prove the Cauchy–Crofton and co-area formulae for the o-minimal Hausdorff measure. We also prove that every definable set can be partitioned into “basic rectifiable sets”, and that the Whitney arc property holds for basic rectifiable sets.

On uniform definability of types over finite sets

AbstractIn this paper, using definability of types over indiscernible sequences as a template, we study a property of formulas and theories called “uniform definability of types over finite sets” (UDTFS). We explore UDTFS and show how it relates to well-known properties in model theory. We recall that stable theories and weakly o-minimal theories have UDTFS and UDTFS implies dependence. We then show that all dp-minimal theories have UDTFS.

Verification of Bounded Discrete Horizon Hybrid Automata

We consider the class of o-minimally definable hybrid automata with a bounded discrete-transition horizon. We show that for every hybrid automata in this class, there exists a bisimulation of finite index, and that the bisimulation quotient can be effectively constructed when the underlying o-minimal theory is decidable. More importantly, we give natural specifications for hybrid automata which ensure the boundedness of discrete-transition horizons. In addition, we show that these specifications are reasonably tight with respect to the decidability of the models and that they can model modern day real-time and embedded systems. As a result, the analysis of several problems for these systems admit effective algorithms. We provide a representative example of a hybrid automaton in this class. Unlike previously examined subclasses of o-minimally defined hybrid automata with decidable verification properties and extended o-minimal hybrid automata, we do not impose re-initialization of the continuous variables in a memoryless fashion when a discrete transition is taken. Our class of hybrid systems has both rich continuous dynamics and strong discrete-continuous coupling, showing that it is not necessary to either simplify the continuous dynamics or restrict the discrete dynamics to achieve decidability.

Weakly one-based geometric theories

AbstractWe study the class of weakly locally modular geometric theories introduced in [4], a common generalization of the classes of linear SU-rank 1 and linear o-minimal theories. We find new conditions equivalent to weak local modularity: “weak one-basedness”, absence of type definable “almost quasidesigns”, and “generic linearity”. Among other things, we show that weak one-basedness is closed under reducts. We also show that the lovely pair expansion of a non-trivial weakly one-basedω-categorical geometric theory interprets an infinite vector space over a finite field.

Finding generically stable measures

AbstractThis work builds on previous papers by Hrushovski, Pillay and the author where Keisler measures over NIP theories are studied. We discuss two constructions for obtaining generically stable measures in this context. First, we show how to symmetrize an arbitrary invariant measure to obtain a generically stable one from it. Next, we show that suitable sigma-additive probability measures give rise to generically stable Keisler measures. Also included is a proof that generically stable measures over o-minimal theories and the p-adics are smooth.

Commutators in groups definable in o-minimal structures

We prove the definability, and actually the finiteness of the commutator width, of many commutator subgroups in groups definable in ominimal structures. It applies in particular to derived series and to lower central series of solvable groups. Along the way, we prove some generalities on groups with the descending chain condition on definable subgroups and/or with a definable and additive dimension.

Embedded o-minimal structures

(Bull. London Math. Soc. 42 (2010) 64–74) There is a serious mistake in the proof of Theorem 1 in the above mentioned paper. Consequently, we must withdraw the claim of having proved that theorem.

Counting special points: Logic, diophantine geometry, and transcendence theory

We expose a theorem of Pila and Wilkie on counting rational points in sets definable in o-minimal structures and some applications of this theorem to problems in diophantine geometry due to Masser, Peterzil, Pila, Starchenko, and

Convergence results for function spaces over o-minimal structures

We begin the development of a theory of Banach spaces in the definable setting of o-minimal structures. We outline several results which develop the theory of compact embeddings for explicitly given function spaces. One aim is to explain the substantive underpinnings of an important observation used in the proof of the Reparameterization Theorem of Pila and Wilkie in (2). We place this observation in the broader context of our theory and demonstrate how it may be refined further. 2000 Mathematics Subject Classification 03C64 (primary); 46B99 (secondary)

Topology of definable abelian groups in o-minimal structures

In this note, we prove that every definably connected, definably compact abelian definable group G in an o-minimal expansion of a real closed field with dim(G)≠4 is definably homeomorphic to a torus of the same dimension. Moreover, in the semialgebraic case the result holds for all dimensions.

Splitting definably compact groups in o-minimal structures

AbstractAn argument of A. Borel [Bor-61, Proposition 3.1] shows that every compact connected Lie group is homeomorphic to the Cartesian product of its derived subgroup and a torus. We prove a parallel result for definably compact definably connected groups definable in an o-minimal expansion of a real closed field. As opposed to the Lie case, however, we provide an example showing that the derived subgroup may not have a definable semidirect complement.

Smooth roots of hyperbolic polynomials with definable coefficients

We prove that the roots of a definable $C^\infty$ curve of monic hyperbolic polynomials admit a definable $C^\infty$ parameterization, where `definable' refers to any fixed o-minimal structure on $(\mathbb R,+,\cdot)$. Moreover, we provide sufficient conditions, in terms of the differentiability of the coefficients and the order of contact of the roots, for the existence of $C^p$ (for $p \in \mathbb N$) arrangements of the roots in both the definable and the non-definable case. These conditions are sharp in the definable and under an additional assumption also in the non-definable case. In particular, we obtain a simple proof of Bronshtein's theorem in the definable setting. We prove that the roots of definable $C^\infty$ curves of complex polynomials can be desingularized by means of local power substitutions $t \mapsto \pm t^N$. For a definable continuous curve of complex polynomials we show that any continuous choice of roots is actually locally absolutely continuous.

Dp-Minimality: Basic Facts and Examples

We study the notion of dp-minimality, beginning by providing several essential facts about dp-minimality, establishing several equivalent definitions for dp-minimality, and comparing dp-minimality to other minimality notions. The majority of the rest of the paper is dedicated to examples. We establish via a simple proof that any weakly o-minimal theory is dp-minimal and then give an example of a weakly o-minimal group not obtained by adding traces of externally definable sets. Next we give an example of a divisible ordered Abelian group which is dp-minimal and not weakly o-minimal. Finally we establish that the field of p-adic numbers is dp-minimal.

John functions for o-minimal domains

Abstract A John function is a continuously differentiable function whose gradient is bounded by the reciprocal of the Euclidean distance to the boundary of the domain. Here we construct John functions whose gradient norms have positive lower bounds and whose domain is a definable set of an o-minimal structure. We prove their definability and give explicit estimates for a number of John functions required for a given set to obtain uniform positive lower bounds.