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

Let Y be a real algebraic subset of ${}^m$ and $F:Y{\to }^n$ be a polynomial map. We show that there exist real polynomial functions $g_1,...,g_s$ on ${}^n$ such that the Euler characteristic of fibres of $F$ is the sum of signs of $g_i$.

The paper is concerned with the relations between real and complex topological invariants of germs of real-analytic functions. We give a formula for the Euler characteristic of the real Milnor fibres of a real-analytic germ in terms of the Milnor numbers of appropriate functions.

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 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.

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, if a meromorphic function of degree at most four on a real algebraic curve of an arbitrary genus has only real critical points, then it is conjugate to a real meromorphic function by a suitable projective automorphism of the image.

We study a constructive method to find an algebraic curve in the real projective plane with a (possibly singular) topological type given in advance. Our method works if the topological model T to be realized has only double singularities and gives an algebraic curve of degree 2N+2K, where N and K are the numbers of double points and connected components of T. This degree is optimal in the sense that for any choice of the numbers N and K there exist models which cannot be realized algebraically with...