Smooth parameterizations of power-subanalytic sets and compositions of Gevrey functions

We show that if $X$ is an $m$ -dimensional definable set in $\mathbb {R}_\text {an}^\text{pow}$ , the structure of real subanalytic sets with real power maps added, then for any positive integer $r$ there exists a $C^{r}$ -parameterization of $X$ consisting of $cr^{m^{3}}$ maps for some constant $c$ . Moreover, these maps are real analytic and this bound is uniform for a definable family.

Defining integer-valued functions in rings of continuous definable functions over a topological field

Let [Formula: see text] be an expansion of either an ordered field [Formula: see text], or a valued field [Formula: see text]. Given a definable set [Formula: see text] let [Formula: see text] be the ring of continuous definable functions from [Formula: see text] to [Formula: see text]. Under very mild assumptions on the geometry of [Formula: see text] and on the structure [Formula: see text], in particular when [Formula: see text] is [Formula: see text]-minimal or [Formula: see text]-minimal, or an expansion of a local field, we prove that the ring of integers [Formula: see text] is interpretable in [Formula: see text]. If [Formula: see text] is [Formula: see text]-minimal and [Formula: see text] is definably connected of pure dimension [Formula: see text], then [Formula: see text] defines the subring [Formula: see text]. If [Formula: see text] is [Formula: see text]-minimal and [Formula: see text] has no isolated points, then there is a discrete ring [Formula: see text] contained in [Formula: see text] and naturally isomorphic to [Formula: see text], such that the ring of functions [Formula: see text] which take values in [Formula: see text] is definable in [Formula: see text].

Lipschitz–Killing curvatures and polar images

We relate the Lipschitz–Killing measures of a definable set X ⊂ ℝn in an o-minimal structure to the volumes of generic polar images. For smooth submanifolds of ℝn, such results were established by Langevin and Shifrin. Then we give infinitesimal versions of these results. As a corollary, we obtain a relation between the polar invariants of Comte and Merle and the densities of generic polar images.

STRONG DENSITY OF DEFINABLE TYPES AND CLOSED ORDERED DIFFERENTIAL FIELDS

The following strong form of density of definable types is introduced for theories T admitting a fibered dimension function d: given a model M of T and a definable set X ⊆ Mn, there is a definable type p in X, definable over a code for X and of the same d-dimension as X. 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.

INTERPRETABLE SETS IN DENSE O-MINIMAL STRUCTURES

We give an example of a dense o-minimal structure in which there is a definable quotient that cannot be eliminated, even after naming parameters. Equivalently, there is an interpretable set which cannot be put in parametrically definable bijection with any definable set. This gives a negative answer to a question of Eleftheriou, Peterzil, and Ramakrishnan. Additionally, we show that interpretable sets in dense o-minimal structures admit definable topologies which are “tame” in several ways: (a) they are Hausdorff, (b) every point has a neighborhood which is definably homeomorphic to a definable set, (c) definable functions are piecewise continuous, (d) definable subsets have finitely many definably connected components, and (e) the frontier of a definable subset has lower dimension than the subset itself.

AN ALGEBRAIC APPROACH TO MSO-DEFINABILITY ON COUNTABLE LINEAR ORDERINGS

We develop an algebraic notion of recognizability for languages of words indexed by countable linear orderings. We prove that this notion is effectively equivalent to definability in monadic second-order (MSO) logic. We also provide three logical applications. First, we establish the first known collapse result for the quantifier alternation of MSO logic over countable linear orderings. Second, we solve an open problem posed by Gurevich and Rabinovich, concerning the MSO-definability of sets of rational numbers using the reals in the background. Third, we establish the MSO-definability of the set of yields induced by an MSO-definable set of trees, confirming a conjecture posed by Bruyère, Carton, and Sénizergues.

A closer look at the stable completion

This chapter introduces the concept of stable completion and provides a concrete representation of unit vector Mathematical Double-Struck Capital A superscript n in terms of spaces of semi-lattices, with particular emphasis on the frontier between the definable and the topological categories. It begins by constructing a topological embedding of unit vector Mathematical Double-Struck Capital A superscript n into the inverse limit of a system of spaces of semi-lattices L(Hsubscript d) endowed with the linear topology, where Hsubscript d are finite-dimensional vector spaces. The description is extended to the projective setting. The linear topology is then related to the one induced by the finite level morphism L(Hsubscript d). The chapter also considers the condition that if a definable set in L(Hsubscript d) is an intersection of relatively compact sets, then it is itself relatively compact.

Preliminaries

This chapter provides some background material on definable sets, definable types, orthogonality to a definable set, and stable domination, especially in the valued field context. It considers more specifically these concepts in the framework of the theory ACVF of algebraically closed valued fields and describes the definable types concentrating on a stable definable V as an ind-definable set. It also proves a key result that demonstrates definable types as integrals of stably dominated types along some definable type on the value group sort. Finally, it discusses the notion of pseudo-Galois coverings. Every nonempty definable set over an algebraically closed substructure of a model of ACVF extends to a definable type.

The main theorem

This chapter introduces the main theorem, which states: Let V be a quasi-projective variety over a valued field F and let X be a definable subset of V x Γ‎superscript Script Small l subscript infinity over some base set V ⊂ VF ∪ Γ‎, with F = VF(A). Then there exists an A-definable deformation retraction h : I × unit vector X → unit vector X with image an iso-definable subset definably homeomorphic to a definable subset of Γ‎superscript w subscript Infinity, for some finite A-definable set w. The chapter presents several preliminary reductions to essentially reduce to a curve fibration. It then constructs a relative curve homotopy and a liftable base homotopy, along with a purely combinatorial homotopy in the Γ‎-world. It also constructs the homotopy retraction by concatenating the previous three homotopies together with an inflation homotopy. Finally, it describes a uniform version of the main theorem with respect to parameters.