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...

We discuss some conditions which guarantee that the Kuratowski limit of a sequence of analytic sets is a Nash set.

Let V be a real algebraic manifold of positive dimension. The aim of this paper is to show that, for every integer b (arbitrarily large), there exists a trivial Nash family $={V_y}_{y\in R^b}$ of real algebraic manifolds such that V₀ = V, is an algebraic family of real algebraic manifolds over $y\in R^b\setminus 0$ (possibly singular over y = 0) and is perfectly parametrized by $R^b$ in the sense that $V_y$ is birationally nonisomorphic to $V_z$ for every $y,z\in R^b$ with y ≠ z. A similar result continues to hold if V is a singular real algebraic set.

We introduce a notion of generic real algebraic variety and we study the space of morphisms into these varieties. Let $Z$ be a real algebraic variety. We say that $Z$ is generic if there exist a finite family ${\left\{D_i\right\}}_{i=1}^n$ of irreducible real algebraic curves with genus $\ge 2$ and a biregular embedding of $Z$ into the product variety ${\prod }_{i=1}^n{D}_i$. A bijective map $\varphi :{\tilde{Z}}^{\phantom{1}}\to Z$ from a real algebraic variety $\tilde{Z}$ to $Z$ is called weak change of the algebraic structure of $Z$ if it is regular and its inverse is a Nash map. Generic real algebraic varieties...