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...

Let (ₙ)ₙ be a quasianalytic differentiable system. Let m ∈ ℕ. We consider the following problem: let $f{\in }_m$ and f̂ be its Taylor series at $0\in {ℝ}^m$. Split the set ${\mathbb{N}}^m$ of exponents into two disjoint subsets A and B, ${\mathbb{N}}^m=A\cup B$, and decompose the formal series f̂ into the sum of two formal series G and H, supported by A and B, respectively. Do there exist $g,h{\in }_m$ with Taylor series at zero G and H, respectively? The main result of this paper is the following: if we have a positive answer to the above problem for some m ≥ 2, then...

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

In this short note we utilize the Borsuk-Ulam Anitpodal Theorem to present a simple proof of the following generalization of the “Ham Sandwich Theorem”: Let $A_1,...,A_m\subseteq {ℝ}^n$ be subsets with finite Lebesgue measure. Then, for any sequence $f_0,...,f_m$ of $ℝ$-linearly independent polynomials in the polynomial ring $ℝ[X_1,...,X_n]$ there are real numbers ${\lambda }_0,...,{\lambda }_m$, not all zero, such that the real affine variety $\{x\in {ℝ}^n;{\lambda }_0{f}_0\left(x\right)+\cdots +{\lambda }_m{f}_m\left(x\right)=0\}$ simultaneously bisects each of subsets $A_k$, $k=1,...,m$. Then some its applications are studied.

Let f: ℝⁿ → ℝ be a C² semialgebraic function and let c be an asymptotic critical value of f. We prove that there exists a smallest rational number ${\varrho }_c\le 1$ such that |x|·|∇f| and ${|f\left(x\right)-c|}^{{\varrho }_c}$ are separated at infinity. If c is a regular value and ${\varrho }_c<1$, then f is a locally trivial fibration over c, and the trivialisation is realised by the flow of the gradient field of f.

We prove the o-minimal generalization of the Łojasiewicz inequality $\parallel \mathrm{grad}f\parallel \ge |f{|}^{\alpha }$, with $\alpha \&lt;1$, 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.

We prove that the number of distinct homotopy types of limits of one-parameter semi-algebraic families of closed and bounded semi-algebraic sets is bounded singly exponentially in the additive complexity of any quantifier-free first order formula defining the family. As an important consequence, we derive that the number of distinct homotopy types of semi-algebraic subsets of ${ℝ}^k$ defined by a quantifier-free first order formula $\Phi$, where the sum of the additive complexities of the polynomials appearing...

We estimate the expected value of the gradient degree of certain Gaussian random polynomials in two variables and discuss its relations with some other numerical invariants of random polynomials