We consider a global semianalytic set defined by real analytic functions definable in an o-minimal structure. When the o-minimal structure is polynomially bounded, we show that the closure of this set is a global semianalytic set defined by definable real analytic functions. We also demonstrate that a connected component of a planar global semianalytic set defined by real analytic functions definable in a substructure of the restricted analytic field is a global semianalytic set defined by definable real analytic functions.
We investigate the Dirichlet solution for a semianalytic continuous function on the boundary of a semianalytic bounded domain in the plane. We show that the germ of the Dirichlet solution at a boundary point with angle greater than 0 lies in a certain quasianalytic class used by Ilyashenko in his work on Hilbert's 16th problem. With this result we can prove that the Dirichlet solution is definable in an o-minimal structure if the angle at a singular boundary point of the domain is an irrational multiple of $π$.
We consider the Riemann Mapping Theorem in the case of a bounded simply connected and semianalytic domain. We show that the germ at 0 of the Riemann map (i.e. biholomorphic map) from the upper half plane to such a domain can be realized in a certain quasianalytic class if the angle of the boundary at the point to which 0 is mapped, is greater than 0. This quasianalytic class was introduced and used by Ilyashenko in his work on Hilbert's 16th problem. With this result we can prove that the Riemann map from a bounded simply connected semianalytic domain onto the unit ball is definable in an o-mi…
The randomization of a complete first order theory T is the complete continuous theory T^R with two sorts, a sort for random elements of models of T, and a sort for events in an underlying probability space. We give necessary and sufficient conditions for an element to be definable over a set of parameters in a model of T^R.
In this paper, we undertake a systematic model and valuation theoretic study of the class of ordered fields which are dense in their real closure. We apply this study to determine definable henselian valuations on ordered fields, in the language of ordered rings. In light of our results, we re-examine the Shelah-Hasson Conjecture (specialised to ordered fields) and provide an example limiting its valuation theoretic conclusions.
We investigate the computability of algebraic closure and definable closure with respect to a collection of formulas. We show that for a computable collection of formulas of quantifier rank at most $n$, in any given computable structure, both algebraic and definable closure with respect to that collection are $Σ^0_{n+2}$ sets. We further show that these bounds are tight.
We study algebraic closure and its relation with definable closure in free groups and more generally in torsion-free hyperbolic groups. Given a torsion-free hyperbolic group G and a nonabelian subgroup A of G, we describe G as a constructible group from the algebraic closure of A along cyclic subgroups. In particular, it follows that the algebraic closure of A is finitely generated, quasiconvex and hyperbolic. Suppose that G is free. Then the definable closure of A is a free factor of the algebraic closure of A and the rank of these groups is bounded by that of G. We prove that the algebraic c…
Brouwer (1927) claimed that every function from the Baire space to natural numbers is induced by a neighbourhood function whose domain admits bar induction. We show that Brouwer's claim is provable in Heyting arithmetic in all finite types ($\mathrm{HA}^ω$) for definable functions of the system. The proof does not rely on elaborate proof theoretic methods such as normalisation or ordinal analysis. Instead, we internalise in $\mathrm{HA}^ω$ the dialogue tree interpretation of Gödel's system T due to Escardó (2013). The interpretation determines a syntactic translation of terms, which yields a n…
We define a discrete closure operation for definably complete locally o-minimal structures $\mathcal M$. The pair of the underlying set of $\mathcal M$ and the discrete closure operation forms a pregeometry. We define the rank of a definable set over a set of parameters using this fact. A definable set $X$ is of dimension equal to the rank of $X$ over the set of parameters of a formula defining the set $X$. The structure $\mathcal M$ is simultaneously a first-order topological structure. The dimension rank of a set definable in the first-order topological structure $\mathcal M$ also coincides …
We consider definably complete and Baire expansions of ordered fields: every definable subset of the domain of the structure has a supremum and the domain can not be written as the union of a definable increasing family of nowhere dense sets. Every expansion of the real field is definably complete and Baire. So is every o-minimal expansion of a field. However, unlike the o-minimal case, the structures considered form an elementary class. In this context we prove a version of Kuratowski-Ulam's Theorem and some restricted version of Sard's Lemma. We use the above results to prove the following v…
In 1979 Schwichtenberg showed that the System $\text{T}$ definable functionals are closed under a rule-like version Spector's bar recursion of lowest type levels $0$ and $1$. More precisely, if the functional $Y$ which controls the stopping condition of Spector's bar recursor is $\text{T}$-definable, then the corresponding bar recursion of type levels $0$ and $1$ is already $\text{T}$-definable. Schwichtenberg's original proof, however, relies on a detour through Tait's infinitary terms and the correspondence between ordinal recursion for $α< \varepsilon_0$ and primitive recursion over finite …
We investigate what henselian valuations on ordered fields are definable in the language of ordered rings. This leads towards a systematic study of the class of ordered fields which are dense in their real closure. Some results have connections to recent conjectures on definability of henselian valuations in strongly NIP fields. Moreover, we obtain a complete characterisation of strongly NIP almost real closed fields.
We give necessary and sufficient conditions on a non-oscillatory curve in an o-minimal field such that, for any bounded definable function, the germ of the function on an initial segment of the curve can be continuously extended to a closed definable set. This situation is translated into a question about types: What are the conditions on an $n$-type such that, for any bounded definable function, there is a definable set containing the type on which the function is continuous, and can be extended continuously to the set's closure? All such types are definable, and we give the precise condition…
On differentiability in weakly o-minimal expansions of fields
University of Pamplona（潘普洛纳大学）；Technion - Israel Institute of Technology（以色列理工学院）；Ben Gurion University of the Negev（内盖夫本-古里安大学）；Fudan University（复旦大学）；University of Haifa（海法大学）
We 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…
Let $A\sub \R^{n+r}$ be a set definable in an o-minimal expansion $§$ 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 Pfaf…
We study definable sets $D$ of SU-rank 1 in $M^{eq}$, where $M$ is a countable homogeneous and simple structure in a language with finite relational vocabulary. Each such $D$ can be seen as a `canonically embedded structure', which inherits all relations on $D$ which are definable in $M^{eq}$, and has no other definable relations. Our results imply that if no relation symbol of the language of $M$ has arity higher than 2, then there is a close relationship between triviality of dependence and $D$ being a reduct of a binary random structure. Somewhat more preciely: (a) if for every $n \geq 2$, …
We study the modal logic of the closure algebra $P_2$, generated by the set of all polygons in the Euclidean plane $\mathbb{R}^2$. We show that this logic is finitely axiomatizable, is complete with respect to the class of frames we call "crown" frames, is not first order definable, does not have the Craig interpolation property, and its validity problem is PSPACE-complete.
The paper is a continuation of our earlier article where we developed a theory of active and non-active infinitesimals and intended to establish quantifier elimination in quasianalytic structures. That article, however, did not attain full generality, which refers to one of its results, namely the theorem on an active infinitesimal, playing an essential role in our non-standard analysis. The general case was covered in our subsequent preprint, which constitutes a basis for the approach presented here. We also provide a quasianalytic exposition of the results concerning rectilinearization of te…
Let R be an o-minimal expansion of the real field. We introduce a class of Hausdorff limits, the T-infinity limits over R, that do not in general fall under the scope of Marker and Steinhorn's definability-of-types theorem. We prove that if R admits analytic cell decomposition, then every T-infinity limit over R is definable in the pfaffian closure of R.
We introduce the notion of $R$-analytic functions. These are definable in an o-minimal expansion of a real closed field $R$ and are locally the restriction of a $K$-differentiable function (defined by Peterzil and Starchenko) where $K=R[\sqrt{-1}]$ is the algebraic closure of $R$. The class of these functions in this general setting exhibits the nice properties of real analytic functions. We also define strongly $R$-analytic functions. These are globally the restriction of a $K$-differentiable function. We show that in arbitrary models of important o-minimal theories strongly $R$-analytic func…
We consider the covering map $π:\mathbb{C}^n\to \mathbb{T}$ of a compact complex torus. Given an algebraic variety $X\subseteq \mathbb{C}^n$ we describe the topological closure of $π(X)$ in $\mathbb T$. We obtain a similar description when $\mathbb{T}$ is a real torus and $X\subseteq \mathbb{R}^n$ is a set definable in an o-minimal structure over the reals.
Let $G$ be a connected, simply connected nilpotent Lie group, identified with a real algebraic subgroup of $\mathrm{UT}(n,\mathbb{R})$, and let $Γ$ be a lattice in $G$, with $π:G\to G/Γ$ the quotient map. For a semi-algebraic $X\subseteq G$, and more generally a definable set in an o-minimal structure on the real field, we consider the topological closure of $π(X)$ in the compact nilmanifold $G/Γ$. Our theorem describes $\mathrm{cl}(π(X))$ in terms of finitely many families of cosets of real algebraic subgroups of $G$. The underlying families are extracted from $X$, independently of $Γ$. We al…
Let $X \subset \mathbb{C}^n$ be an algebraic variety, and let $Λ\subset \mathbb{C}^n$ be a discrete subgroup whose real and complex spans agree. We describe the topological closure of the image of $X$ in $\mathbb{C}^n / Λ$, thereby extending a result of Peterzil-Starchenko in the case when $Λ$ is cocompact. We also obtain a similar extension when $X\subset \mathbb{R}^n$ is definable in an o-minimal structure with no restrictions on $Λ$, and as an application prove the following conjecture of Gallinaro: for a closed semi-algebraic $X\subset \mathbb{C}^n$ (such as a complex algebraic variety) an…
Let L be a countable language. We say that a countable infinite L-structure M admits an invariant measure when there is a probability measure on the space of L-structures with the same underlying set as M that is invariant under permutations of that set, and that assigns measure one to the isomorphism class of M. We show that M admits an invariant measure if and only if it has trivial definable closure, i.e., the pointwise stabilizer in Aut(M) of an arbitrary finite tuple of M fixes no additional points. When M is a Fraisse limit in a relational language, this amounts to requiring that the age…
