O-minimal Research Articles

Tempered distributions and Schwartz functions on definable manifolds

We define the spaces of Schwartz functions, tempered functions and tempered distributions on manifolds definable in polynomially bounded o-minimal structures. We show that all the classical properties that these spaces have in the Nash category, as first studied in Fokko du Cloux's work, also hold in this generalized setting. We also show that on manifolds definable in o-minimal structures that are not polynomially bounded, such a theory can not be constructed. We present some possible applications, mainly in representation theory.

Theorem of Lion and o-minimal structures which do not admit $\mathcal {C}^{\infty }$-cell decompositions

Theorem of Lion and o-minimal structures which do not admit $\mathcal {C}^{\infty }$-cell decompositions

Deformation retracts to intersections of Whitney stratifications

We give a counterexample to a conjecture of Eyral on the existence of deformation retracts to intersections of Whitney stratifications embedded in a smooth manifold. We then prove that the conjecture holds if the stratifications are definable in some o-minimal structure without assuming any regularity conditions. Moreover, we also show that the conjecture holds for Whitney stratifications if they intersect transversally.

Pairs of theories satisfying a Mordell–Lang condition

This paper proposes a new setup for studying pairs of structures. This new framework includes many of the previously studied classes of pairs, such as dense pairs of o-minimal structures, lovely pairs, fields with Mann groups, and $H$-structures, but also includes new ones, such as pairs consisting of a real closed field and a pseudo real closed subfield, and pairs of vector spaces with different fields of scalars. We use the larger generality of this framework to answer three concrete open questions raised in earlier work on this subject.

Uniformly locally o-minimal structures and locally o-minimal structures admitting local definable cell decomposition

We define and investigate a uniformly locally o-minimal structure of the second kind in this paper. All uniformly locally o-minimal structures of the second kind have local monotonicity, which is a local version of monotonicity theorem of o-minimal structures. We also demonstrate a local definable cell decomposition theorem for definably complete uniformly locally o-minimal structures of the second kind. We define dimension of a definable set and investigate its basic properties when the given structure is a locally o-minimal structure which admits local definable cell decomposition.

Expansions of the Real Field by Canonical Products

AbstractWe consider expansions of o-minimal structures on the real field by collections of restrictions to the positive real line of the canonical Weierstrass products associated with sequences such as$(-n^{s})_{n>0}$(for$s>0$) and$(-s^{n})_{n>0}$(for$s>1$), and also expansions by associated functions such as logarithmic derivatives. There are only three possible outcomes known so far: (i) the expansion is o-minimal (that is, definable sets have only finitely many connected components); (ii) every Borel subset of each$\mathbb{R}^{n}$is definable; (iii) the expansion is interdefinable with a structure of the form$(\mathfrak{R}^{\prime },\unicode[STIX]{x1D6FC}^{\mathbb{Z}})$where$\unicode[STIX]{x1D6FC}>1$,$\unicode[STIX]{x1D6FC}^{\mathbb{Z}}$is the set of all integer powers of$\unicode[STIX]{x1D6FC}$, and$\mathfrak{R}^{\prime }$is o-minimal and defines no irrational power functions.

CHARACTERIZING O-MINIMAL GROUPS IN TAME EXPANSIONS OF O-MINIMAL STRUCTURES

AbstractWe establish the first global results for groups definable in tame expansions of o-minimal structures. Let ${\mathcal{N}}$ be an expansion of an o-minimal structure ${\mathcal{M}}$ that admits a good dimension theory. The setting includes dense pairs of o-minimal structures, expansions of ${\mathcal{M}}$ by a Mann group, or by a subgroup of an elliptic curve, or a dense independent set. We prove: (1) a Weil’s group chunk theorem that guarantees a definable group with an o-minimal group chunk is o-minimal, (2) a full characterization of those definable groups that are o-minimal as those groups that have maximal dimension; namely, their dimension equals the dimension of their topological closure, (3) as an application, if ${\mathcal{N}}$ expands ${\mathcal{M}}$ by a dense independent set, then every definable group is o-minimal.

Small sets in dense pairs

Let $$\widetilde{\cal M} = \langle {\cal M},P\rangle $$ be an expansion of an o-minimal structure $$\mathcal M$$ by a dense set P ⊆ M, such that three tameness conditions hold. We prove that the induced structure on P by $$\mathcal M$$ eliminates imaginaries. As a corollary, we obtain that every small set X definable in $$\widetilde{\cal M}$$ can be definably embedded into some Pl, uniformly in parameters, settling a question from [8]. We verify the tameness conditions in three examples: dense pairs of real closed fields, expansions of ℳ by a dense independent set, and expansions by a dense divisible multiplicative group with the Mann property. Along the way, we point out a gap in the proof of a relevant elimination of imaginaries result in Wencel [13]. The above results are in contrast to recent literature, as it is known in general that $$\widetilde{\cal M}$$ does not eliminate imaginaries, and neither it nor the induced structure on P admits definable Skolem functions.

Definable one-dimensional topologies in O-minimal structures

We consider definable topological spaces of dimension one in o-minimal structures, and state several equivalent conditions for when such a topological space $$\left( X,\tau \right) $$ is definably homeomorphic to an affine definable space (namely, a definable subset of $$M^{n}$$ with the induced subspace topology). One of the main results says that it is sufficient for X to be regular and decompose into finitely many definably connected components.

On expansions of models of weakly o-minimal theories by binary predicates

On expansions of models of weakly o-minimal theories by binary predicates

Whitney’s extension problem in o-minimal structures

In 1934, H. Whitney asked how one can determine whether a real-valued function on a closed subset of \mathbb{R}^n is the restriction of a C^m -function on \mathbb{R}^n . A complete answer to this question was found much later by C. Fefferman in the early 2000s. Here, we work in an o-minimal expansion of a real closed field and solve the C^1 -case of Whitney's extension problem in this context. Our main tool is a definable version of Michael's selection theorem, and we include other another application of this theorem, to solving linear equations in the ring of definable continuous functions.

Maximality of the Countable Spectrum in Small Quite o-Minimal Theories

We give a criterion for the countable spectrum to be maximal in small binary quite o-minimal theories of finite convexity rank.

Cartan subgroups and regular points of o‐minimal groups

Let G be a group definable in an o-minimal structure M. We prove that the union of the Cartan subgroups of G is a dense subset of G. When M is an expansion of a real closed field we give a characterization of Cartan subgroups of G via their Lie algebras which allow us to prove firstly, that every Cartan subalgebra of the Lie algebra of G is the Lie algebra of a definable subgroup - a Cartan subgroup of G -, and secondly, that the set of regular points of G - a dense subset of G - is formed by points which belong to a unique Cartan subgroup of G.

The derived subgroup of linear and simply-connected o-minimal groups

We show that the derived subgroup of a linear definable group in an o-minimal structure is also definable, extending the semialgebraic case proved in [18]. We also show the definability of the derived subgroup in case that the group is simply-connected.

On the set of local extrema of a subanalytic function

Let $${\mathfrak {F}}$$ be a category of subanalytic subsets of real analytic manifolds that is closed under basic set-theoretical operations (locally finite unions, difference and product) and basic topological operations (taking connected components and closures). Let M be a real analytic manifold and denote $${\mathfrak {F}}(M)$$ the family of the subsets of M that belong to the category $${\mathfrak {F}}$$. Let $$f:X\rightarrow \mathbb {R}$$ be a subanalytic function on a subset $$X\in {\mathfrak {F}}(M)$$ such that the inverse image under f of each interval of $$\mathbb {R}$$ belongs to $${\mathfrak {F}}(M)$$. Let $$\mathrm{Max}(f)$$ be the set of local maxima of f and consider its level sets $$\mathrm{Max}_\lambda (f):=\mathrm{Max}(f)\cap \{f=\lambda \}=\{f=\lambda \}{\setminus }{\text {Cl}}(\{f>\lambda \})$$ for each $$\lambda \in \mathbb {R}$$. In this work we show that if f is continuous, then $$\mathrm{Max}(f)=\bigsqcup _{\lambda \in \mathbb {R}}\mathrm{Max}_\lambda (f)\in {\mathfrak {F}}(M)$$ if and only if the family $$\{\mathrm{Max}_\lambda (f)\}_{\lambda \in \mathbb {R}}$$ is locally finite in M. If we erase continuity condition, there exist subanalytic functions $$f:X\rightarrow M$$ such that $$\mathrm{Max}(f)\in {\mathfrak {F}}(M)$$, but the family $$\{\mathrm{Max}_\lambda (f)\}_{\lambda \in \mathbb {R}}$$ is not locally finite in M or such that $$\mathrm{Max}(f)$$ is connected but it is not even subanalytic. We show in addition that if $${\mathfrak {F}}$$ is the category of subanalytic sets and $$f:X\rightarrow \mathbb {R}$$ is a (non-necessarily continuous) subanalytic map f that maps relatively compact subsets of M contained in X to bounded subsets of $$\mathbb {R}$$, then $$\mathrm{Max}(f)\in {\mathfrak {F}}(M)$$ and the family $$\{\mathrm{Max}_\lambda (f)\}_{\lambda \in \mathbb {R}}$$ is locally finite in M. An example of this type of functions are continuous subanalytic functions on closed subanalytic subsets of M. The previous results imply that if $${\mathfrak {F}}$$ is either the category of semianalytic sets or the category of C-semianalytic sets and f is the restriction to an $${\mathfrak {F}}$$-subset of M of an analytic function on M, then the family $$\{\mathrm{Max}_\lambda (f)\}_{\lambda \in \mathbb {R}}$$ is locally finite in M and $$\mathrm{Max}(f)=\bigsqcup _{\lambda \in \mathbb {R}}\mathrm{Max}_\lambda (f)\in {\mathfrak {F}}(M)$$. We also show that if the category $${\mathfrak {F}}$$ contains the intersections of algebraic sets with real analytic submanifolds and $$X\in {\mathfrak {F}}(M)$$ is not closed in M, then there exists a continuous subanalytic function $$f:X\rightarrow \mathbb {R}$$ with graph belonging to $${\mathfrak {F}}(M\times \mathbb {R})$$ such that inverse images under f of the intervals of $$\mathbb {R}$$ belong to $${\mathfrak {F}}(M)$$ but $$\mathrm{Max}(f)$$ does not belong to $${\mathfrak {F}}(M)$$. As subanalytic sets are locally connected, the set of non-openness points of a continuous subanalytic function $$f:X\rightarrow \mathbb {R}$$ coincides with the set of local extrema $$\mathrm{Extr}(f):=\mathrm{Max}(f)\cup \mathrm{Min}(f)$$. This means that if $$f:X\rightarrow \mathbb {R}$$ is a continuous subanalytic function defined on a closed set $$X\in {\mathfrak {F}}(M)$$ such that the inverse image under f of each interval of $$\mathbb {R}$$ belongs to $${\mathfrak {F}}(M)$$, then the set $$\mathrm{Op}(f)$$ of openness points of f belongs to $${\mathfrak {F}}(M)$$. Again the closedness of X in M is crucial to guarantee that $$\mathrm{Op}(f)$$ belongs to $${\mathfrak {F}}(M)$$. The type of results stated above are straightforward if $${\mathfrak {F}}$$ is an o-minimal structure of subanalytic sets. However, the proof of the previous results requires further work for a category $${\mathfrak {F}}$$ of subanalytic sets that does not constitute an o-minimal structure.

STRONG DENSITY OF DEFINABLE TYPES AND CLOSED ORDERED DIFFERENTIAL FIELDS

AbstractThe following strong form of density of definable types is introduced for theoriesTadmitting a fibered dimension functiond: given a modelMofTand a definable setX⊆Mn, there is a definable typepinX, definable over a code forXand of the samed-dimension asX. Both o-minimal theories and the theory of closed ordered differential fields (CODF) are shown to have this property. As an application, we derive a new proof of elimination of imaginaries for CODF.

A THEORY OF PAIRS FOR NON-VALUATIONAL STRUCTURES

AbstractGiven a weakly o-minimal structure${\cal M}$and its o-minimal completion$\bar{{\cal M}}$, we first associate to$\bar{{\cal M}}$a canonical language and then prove thatTh$\left( {\cal M} \right)$determines$Th\left( {\bar{{\cal M}}} \right)$. We then investigate the theory of the pair$\left( {\bar{{\cal M}},{\cal M}} \right)$in the spirit of the theory of dense pairs of o-minimal structures, and prove, among other results, that it is near model complete, and every definable open subset of${\bar{M}^n}$is already definable in$\bar{{\cal M}}$.We give an example of a weakly o-minimal structure interpreting$\bar{{\cal M}}$and show that it is not elementarily equivalent to any reduct of an o-minimal trace.

Transversality of smooth definable maps in O-minimal structures

AbstractWe present a definable smooth version of the Thom transversality theorem. We show further that the set of non-transverse definable smooth maps is nowhere dense in the definable smooth topology. Finally, we prove a definable version of a theorem of Trotman which says that the Whitney (a)-regularity of a stratification is necessary and sufficient for the stability of transversality.

-parametrization in O-minimal Structures

AbstractWe give a geometric and elementary proof of the uniform$\mathscr{C}^{p}$-parametrization theorem of Yomdin and Gromov in arbitrary o-minimal structures.

Algebras of Distributions of Binary Isolating Formulas for Quite o-Minimal Theories

Algebras of distributions of binary isolating formulas over a type for quite o-minimal theories with nonmaximal number of countable models are described. It is proved that an isomorphism of these algebras for two 1-types is characterized by the coincidence of convexity ranks and also by simultaneous satisfaction of isolation, quasirationality, or irrationality of those types. It is shown that for quite o-minimal theories with nonmaximum many countable models, every algebra of distributions of binary isolating formulas over a pair of nonweakly orthogonal types is a generalized commutative monoid.