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.

Let V be a paracompact connected real analytic manifold of dimension 1 or 2, i.e. a smooth curve or surface. We consider it as a subset of some complex analytic manifold VC of the same dimension. Moreover by a prime divisor of V we shall mean the irreducible germ along V of a codimension one subvariety of VC which is an invariant of the complex conjugation. This notion is independent of the choice of the complexification VC. In the one-dimensional case prime divisors are just points, in the two-dimensional...