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.

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

We present a complex analytic proof of the Pila–Wilkie theorem for subanalytic sets. In particular, we replace the use of $C^{r}$-smooth parametrizations by a variant of Weierstrass division. As a consequence we are able to apply the Bombieri–Pila determinant method directly to analytic families without limiting the order of smoothness by a $C^{r}$ parametrization. This technique provides the key inductive step for our recent proof (in a closely related preprint) of the Wilkie conjecture for sets definable using restricted elementary functions. As an illustration of our approach we prove that the rational points of height $H$ in a compact piece of a complex-analytic set of dimension $k$ in $\mathbb{C}^{m}$ are contained in $O(1)$ complex-algebraic hypersurfaces of degree $(\log H)^{k/(m-k)}$. This is a complex-analytic analog of a recent result of Cluckers, Pila, and Wilkie for real subanalytic sets.

Consider quasianalytic local rings of germs of smooth functions closed under composition, implicit equation, and monomial division. We show that if the Weierstrass Preparation Theoremholds in such a ring, then all elements of it are germs of analytic functions.

Let $X$ be a subset in $[-1,1]^{n_0}\,{\subset}\,\Real^{n_0}$ defined by the formula \[ X=\{ {\bf x}_0\,{\mid}\,Q_1{\bf x}_1 Q_2{\bf x}_2 \cdots Q_{\nu}{\bf x}_{\nu} (({\bf x}_0,{\bf x}_1,\,{\ldots}\,,{\bf x}_{\nu}) \in X_{\nu})\}, \] where $Q_i \in \{ \exists, \forall \}$, $Q_i \neq Q_{i{+}1}$, ${\bf x}_i \in \Real^{n_i}$, and $X_{\nu}$ may be either an open or a closed set in $[-1,1]^{n_0+ \cdots +n_{\nu}}{\!}$, being the difference between a finite CW-complex and its subcomplex. An upper bound on each Betti number of $X$ is expressed via a sum of Betti numbers of some sets defined by quantifier-free formulae involving $X_{\nu}$.

In important particular cases of semialgebraic and semi-Pfaffian sets defined by quantifier-free formulae with polynomials and Pfaffian functions respectively, upper bounds on Betti numbers of $X_{\nu}$ are well known. The results allow to extend the bounds to sets defined with quantifiers, in particular to sub-Pfaffian sets.

Let Xℝ ⊂ ℝN a real analytic set such that its complexification Xℂ ⊂ ℂN is normal with an isolated singularity at 0. Let fℝ : Xℝ → ℝ a real analytic function such that its complexification fℂ : Xℂ → ℂ has an isolated singularity at 0 in Xℂ. Assuming an orientation given on to a connected component A of we associate a compact cycle Γ(A) in the Milnor fiber of fℂ which determines completely the poles of the meromorphic extension of or equivalently the asymptotics when T → ±∞ of the oscillating integrals . A topological construction of Γ(A) is given. This completes the results of [BM] paragraph 6.