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.

By an influential theorem of Boman, a function $f$ on an open set $U$ in $\mathbb{R}^{d}$ is smooth (${\mathcal{C}}^{\infty }$) if and only if it is arc-smooth, that is, $f\,\circ \,c$ is smooth for every smooth curve $c:\mathbb{R}\rightarrow U$. In this paper we investigate the validity of this result on closed sets. Our main focus is on sets which are the closure of their interior, so-called fat sets. We obtain an analogue of Boman’s theorem on fat closed sets with Hölder boundary and on fat closed subanalytic sets with the property that every boundary point has a basis of neighborhoods each of which intersects the interior in a connected set. If $X\subseteq \mathbb{R}^{d}$ is any such set and $f:X\rightarrow \mathbb{R}$ is arc-smooth, then $f$ extends to a smooth function defined on $\mathbb{R}^{d}$. We also get a version of the Bochnak–Siciak theorem on all closed fat subanalytic sets and all closed sets with Hölder boundary: if $f:X\rightarrow \mathbb{R}$ is the restriction of a smooth function on $\mathbb{R}^{d}$ which is real analytic along all real analytic curves in $X$, then $f$ extends to a holomorphic function on a neighborhood of $X$ in $\mathbb{C}^{d}$. Similar results hold for non-quasianalytic Denjoy–Carleman classes (of Roumieu type). We will also discuss sharpness and applications of these results.

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

This article presents two constructions motivated by a conjecture of van den Dries and Miller concerning the restricted analytic field with exponentiation. The first construction provides an example of two o-minimal expansions of a real closed field that possess the same field of germs at infinity of one-variable functions and yet define different global one-variable functions. The second construction gives an example of a family of infinitely many distinct maximal polynomially bounded reducts (all this in the sense of definability) of the restricted analytic field with exponentiation.

Given a real analytic set $X$ in a complex manifold and a positive integer $d$, denote by ${{\mathcal{A}}^{d}}$ the set of points $p$ in $X$ at which there exists a germ of a complex analytic set of dimension $d$ contained in $X$. It is proved that ${{\mathcal{A}}^{d}}$ is a closed semianalytic subset of $X$.

In this paper we present Thom’s transversality theorem in o-minimal structures (a generalization of semialgebraic and subanalytic geometry). There are no restrictions on the differentiability class and the dimensions of manifolds involved in comparison withthe general case.

We obtain a uniform linear bound for the Chevalley function at a point in the source of an analytic mapping that is regular in the sense of Gabrielov. There is a version of Chevalley’s lemma also along a fibre, or at a point of the image of a proper analytic mapping. We get a uniform linear bound for the Chevalley function of a closed Nash (or formally Nash) subanalytic set.

Let $K$ be the (real closed) field of Puiseux series in $t$ over ${\mathbf R}$ endowed with the natural linear order. Then the elements of the formal power series rings ${\mathbf R}[\![\xi_1,\dots,\xi_n]\!]$ converge $t$-adically on $[-t,t]^n$, and hence define functions $[-t,t]^n\to K$. Let ${\mathcal L}$ be the language of ordered fields, enriched with symbols for these functions. By Corollary 3.15, $K$ is o-minimal in ${\mathcal L}$. This result is obtained from a quantifier elimination theorem. The proofs use methods from non-Archimedean analysis.

We prove a generalization to the context of real geometry of an intersection formula for the vanishing cycle functor, which in the complex context is due to Dubson, Lê, Ginsburg and Sabbah (after a conjecture of Deligne). It is also a generalization of similar results of Kashiwara and Schapira, where these authors work with a suitable assumption about the micro-support of the corresponding constructible complex of sheaves. We only use a similar assumption about the support of the corresponding characteristic cycle so that our result can be formulated in the language of constructible functions and Lagrangian cycles.

Let $G$ be a real Lie group and $X$ a real analytic manifold. Suppose that $G$ acts analytically on $X$ with finitely many orbits. Then the orbits are subanalytic in $X$. As a consequence we show that the micro-support of a $G$-equivariant sheaf on $X$ is contained in the conormal variety of the $G$-action.