Nous définissons l’espace des germes d’arcs réels tracés sur un ensemble semi-algébrique de ${ℝ}^n$, analogue réel de la théorie développée par Denef et Loeser concernant l’espace des germes d’arcs tracés sur une variété algébrique complexe. Puis, reprenant leur méthodes, nous prouvons la rationalité de la série de Poincaré associée à un ensemble semi-algébrique.

In Example 1, we describe a subset X of the plane and a function on X which has a ${}^k$-extension to the whole ${ℝ}^2$ for each finite, but has no ${}^{\infty }$-extension to ${ℝ}^2$. In Example 2, we construct a similar example of a subanalytic subset of ${ℝ}^5$; much more sophisticated than the first one. The dimensions given here are smallest possible.

Let K be a subfield of the real field, D ⊆ K be a discrete set and f: Dⁿ → K be such that f(Dⁿ) is somewhere dense. Then (K,f) defines ℤ. We present several applications of this result. We show that K expanded by predicates for different cyclic multiplicative subgroups defines ℤ. Moreover, we prove that every definably complete expansion of a subfield of the real field satisfies an analogue of the Baire category theorem.

Let f: ℝⁿ → ℝ be a polynomial function of degree d with f(0) = 0 and ∇f(0) = 0. Łojasiewicz’s gradient inequality states that there exist C > 0 and ϱ ∈ (0,1) such that ${\left|\nabla f\right|\ge C\left|f\right|}^{\varrho }$ in a neighbourhood of the origin. We prove that the smallest such exponent ϱ is not greater than $1-R{(n,d)}^{-1}$ with $R(n,d)=d{(3d-3)}^{n-1}$.

We prove the rationality of the Łojasiewicz exponent for p-adic semi-algebraic functions without compactness hypothesis. In the parametric case, we show that the parameter space can be divided into a finite number of semi-algebraic sets on each of which the Łojasiewicz exponent is constant.

We prove the rationality of the Łojasiewicz exponent for semialgebraic functions without compactness hypothesis. In the parametric situation, we show that the parameter space can be divided into a finite number of semialgebraic sets on each of which the Łojasiewicz exponent is constant.

We study the extensibility of piecewise polynomial functions defined on closed subsets of ${ℝ}^2$ to all of ${ℝ}^2$. The compact subsets of ${ℝ}^2$ on which every piecewise polynomial function is extensible to ${ℝ}^2$ can be characterized in terms of local quasi-convexity if they are definable in an o-minimal expansion of $ℝ$. Even the noncompact closed definable subsets can be characterized if semialgebraic function germs at infinity are dense in the Hardy field of definable germs. We also present a piecewise polynomial...

Let $h:{ℝ}^n\to ℝ$ be a continuous, piecewise-polynomial function. The Pierce-Birkhoff conjecture (1956) is that any such $h$ is representable in the form ${sup}_i{inf}_j{f}_{ij}$, for some finite collection of polynomials $f_{ij}\in ℝ[x_1,...,x_n]$. (A simple example is $h\left(x_1\right)=\left|x_1\right|=sup\{x_1,-x_1\}$.) In 1984, L. Mahé and, independently, G. Efroymson, proved this for $n\le 2$; it remains open for $n\ge 3$. In this paper we prove an analogous result for “generalized polynomials” (also known as signomials), i.e., where the exponents are allowed to be arbitrary real numbers, and not just natural numbers;...