Sia $f:{\mathbb{R}}^m\to {\mathbb{R}}^k$ con $m\ge k\ge 1$ una funzione analitica. Se il luogo critico di $f$ è compatto, esiste una fibrazione $C^{\mathrm{\infty }}$ localmente triviale associata ai livelli $f$. Supponiamo $k\ge 2$ e sia ${\pi }_k$ la proiezione $\left(x_1,\mathrm{\dots },x_{k-1},x_k\right)\mapsto \left(x_1,\mathrm{\dots },x_{k-1},x_k\right)$. Sotto una condizione sul luogo critico di $\tilde{f}={\pi }_k\circ f$ esiste anche una fibrazione $C^{\mathrm{\infty }}$ localmente triviale associata ai livelli di $\tilde{f}$. Siano $F$ e $\tilde{F}$ le fibre rispettitive, e $I$ l'intervallo unità reale. Dimostriamo qui che $\tilde{F}$ è omeomorfa al prodotto $F\times I$. Nel caso di polinomi studiamo criteri effettivi. Diamo inoltre un'applicazione del risultato...

We consider the 17th Hilbert Problem for global real analytic functions in a modified form that involves infinite sums of squares. Then we prove a local-global principle for a real global analytic function to be a sum of squares of global real meromorphic functions. We deduce that an affirmative solution to the 17th Hilbert Problem for global real analytic functions implies the finiteness of the Pythagoras number of the field of global real meromorphic functions, hence that of the field of real...

We show that there is a large class of nonspecial effective divisors of relatively small degree on real algebraic curves having many real components i.e. on M-curves. We apply to 1. complete linear systems on M-curves containing divisors with entirely real support, and 2. morphisms of M-curves into P1.

We associate to a given polynomial map from ${ℂ}^2$ to itself with nonvanishing Jacobian a variety whose homology or intersection homology describes the geometry of singularities at infinity of this map.

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

Let f: ℝⁿ → ℝ be a nonconstant polynomial function. Using the information from the "curve of tangency" of f, we provide a method to determine the Łojasiewicz exponent at infinity of f. As a corollary, we give a computational criterion to decide if the Łojasiewicz exponent at infinity is finite or not. Then we obtain a formula to calculate the set of points at which the polynomial f is not proper. Moreover, a relation between the Łojasiewicz exponent at infinity of f and the problem of computing...

We will prove that the Pierce-Birkhoff Conjecture holds for non-singular two-dimensional affine real algebraic varieties over real closed fields, i.e., if $W$ is such a variety, then every piecewise polynomial function on $W$ can be written as suprema of infima of polynomial functions on $W$. More precisely, we will give a proof of the so-called Connectedness Conjecture for the coordinate rings of such varieties, which implies the Pierce-Birkhoff Conjecture.

Given integers $D\>0,n\>1,0\<r\<n$ and a constant $C\>0$, consider the space of $r$-tuples $\overrightarrow{P}=(P_1...P_r)$ of real polynomials in $n$ variables of degree $\le D$, whose coefficients are $\le C$ in absolute value, and satisfying $\det {\left(\frac{\partial P_i}{\partial x_i}\left(0\right)\right)}_{1\le i,j\le r}=1$. We study the family $\left\{f\right|V\}$ of algebraic functions, where $f$ is a polynomial, and $V=\left\{\right|x|\le \delta ,\overrightarrow{P}(x)=0\},\delta \>0$ being a constant depending only on $n,D,C$. The main result is a quantitative extension theorem for these functions which is uniform in $\overrightarrow{P}$. This is used to prove Bernstein-type inequalities which are again uniform with respect to $\overrightarrow{P}$.The proof is based on...

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 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.