On définit, pour un germe d’ensemble sous-analytique, deux nouvelles suites finies d’invariants numériques. La première a pour termes les localisations des courbures de Lipschitz-Killing classiques, la seconde est l’équivalent réel des caractéristiques évanescentes complexes introduites par M. Kashiwara. On montre que chaque terme d’une de ces suites est combinaison linéaire des termes de l’autre, puis on relie ces invariants à la géométrie des discriminants des projections du germe sur des plans...

Letg:U→ℝ (U open in ℝn) be an analytic and K-subanalytic (i. e. definable in ℝanK, whereK, the field of exponents, is any subfield ofℝ) function. Then the set of points, denoted Σ, whereg does not admit an analytic extension is K-subanalytic andg can be extended analytically to a neighbourhood of Ū.

We extend a result of M. Tamm as follows:Let $f:A\to ℝ,A\subseteq {ℝ}^{m+n}$, be definable in the ordered field of real numbers augmented by all real analytic functions on compact boxes and all power functions $x\mapsto x^r:(0,\infty )\to ℝ,r\in ℝ$. Then there exists $N\in \mathbb{N}$ such that for all $(a,b)\in A$, if $y\mapsto f(a,y)$ is $C^N$ in a neighborhood of $b$, then $y\mapsto f(a,y)$ is real analytic in a neighborhood of $b$.

By an open neighbourhood in ℂⁿ of an open subset Ω of ℝⁿ we mean an open subset Ω' of ℂⁿ such that ℝⁿ ∩ Ω' = Ω. A well known result of H. Grauert implies that any open subset of ℝⁿ admits a fundamental system of Stein open neighbourhoods in ℂⁿ. Another way to state this property is to say that each open subset of ℝⁿ is Stein. We shall prove a similar result in the subanalytic category: every subanalytic open subset in a paracompact real analytic manifold M admits a fundamental system of subanalytic...

In the 70’s, Nekhorochev proved that for an analytic nearly integrable Hamiltonian system, the action variables of the unperturbed Hamiltonian remain nearly constant over an exponentially long time with respect to the size of the perturbation, provided that the unperturbed Hamiltonian satisfies some generic transversality condition known as steepness. Using theorems of real subanalytic geometry, we derive a geometric criterion for steepness: a numerical function $h$ which is real analytic around a...

We prove rectilinearization and uniformization theorems for K-subanalytic (∝anK-definable) sets and functions using the Lion-Rolin formula. Parallel reasoning gives standard results for the subanalytic case.

We consider some variants of Łojasiewicz inequalities for the class of subsets of Euclidean spaces definable from addition, multiplication and exponentiation : Łojasiewicz-type inequalities, global Łojasiewicz inequalities with or without parameters. The rationality of Łojasiewicz’s exponents for this class is also proved.

We show that if X, Y are smooth, compact k-dimensional submanifolds of ℝⁿ and 2k+2 ≤ n, then each diffeomorphism ϕ: X → Y can be extended to a diffeomorphism Φ: ℝⁿ → ℝⁿ which is tame (to be defined in this paper). Moreover, if X, Y are real analytic manifolds and the mapping ϕ is analytic, then we can choose Φ to be also analytic. We extend this result to some interesting categories of closed (not necessarily compact) subsets of ℝⁿ, namely, to the category of Nash submanifolds...

Let $M\subset {ℝ}^n$ be a real-analytic submanifold and H(M) the algebra of real analytic functions on M. If K ⊂ M is a compact subset we consider $S_K=f\in H\left(M\right)\left|f\right(x)\ne 0forallx\in K$; $S_K$ is a multiplicative subset of $H\left(M\right)$. Let $S_K^{-1}H\left(M\right)$ be the localization of H(M) with respect to $S_K$. In this paper we prove, first, that $S_K^{-1}H\left(M\right)$ is a regular ring (hence noetherian) and use this result in two situations:    1) For each open subset $\Omega \subset {ℝ}^n$, we denote by O(Ω) the subalgebra of H(Ω) defined as follows: f ∈ O(Ω) if and only if for all x ∈ Ω, the germ of f at x, $f_x$, is algebraic...

This is a generalized and improved version of our earlier article [Studia Math. 124 (1997)] on the Whitney extension theorem for subanalytic ${}^p$-Whitney fields (with p finite). In this new version we consider Whitney fields definable in an arbitrary o-minimal structure on any real closed field R and obtain an extension which is a ${}^p$-function definable in the same o-minimal structure. The Whitney fields that we consider are defined on any locally closed definable subset of Rⁿ. In such a way, a local...

This paper presents certain characterizations through blowing up of arc-analytic functions definable by a convergent Weierstrass system closed under complexification.

We prove the o-minimal generalization of the Łojasiewicz inequality $\parallel \mathrm{grad}f\parallel \ge |f{|}^{\alpha }$, with $\alpha \<1$, in a neighborhood of $a$, where $f$ is real analytic at $a$ and $f\left(a\right)=0$. We deduce, as in the analytic case, that trajectories of the gradient of a function definable in an o-minimal structure are of uniformly bounded length. We obtain also that the gradient flow gives a retraction onto levels of such functions.

The purpose of this paper is to carry over to the o-minimal settings some results about the Euler characteristic of algebraic and analytic sets. Consider a polynomially bounded o-minimal structure on the field ℝ of reals. A ($C^{\infty }$) smooth definable function φ: U → ℝ on an open set U in ℝⁿ determines two closed subsets W := u ∈ U: φ(u) ≤ 0, Z := u ∈ U: φ(u) = 0. We shall investigate the links of the sets W and Z at the points u ∈ U, which are well defined up to a definable homeomorphism. It is proven...

A local-global version of the implicit function theorem in o-minimal structures and a generalization of the theorem of Wilkie on covering open sets by open cells are proven.

We show that (i) the Pythagoras number of a real analytic set germ is the supremum of the Pythagoras numbers of the curve germs it contains, and (ii) every real analytic curve germ is contained in a real analytic surface germ with the same Pythagoras number (or Pythagoras number $2$ if the curve is Pythagorean). This gives new examples and counterexamples concerning sums of squares and positive semidefinite analytic function germs.