The aim of this series of papers is to develop the theory of minimal models for real algebraic threefolds. The ultimate aim is to understand the topology of the set of real points of real algebraic threefolds. We pay special attention to 3–folds which are birational to projective space and, more generally, to 3–folds of Kodaira dimension minus infinity.present work contains the beginning steps of this program. First we classify 3–dimensional terminal singularities over any field of characteristic...

Existence of loops for non-injective regular analytic transformations of the real plane is shown. As an application, a criterion for injectivity of a regular analytic transformation of ${ℝ}^2$ in terms of the Jacobian and the first and second order partial derivatives is obtained. This criterion is new even in the special case of polynomial transformations.

It is known that for $C^{\infty }$ determining sets Markov’s property is equivalent to Bernstein’s property. We are interested in finding a generalization of this fact for sets which are not $C^{\infty }$ determining. In this paper we give examples of sets which are not $C^{\infty }$ determining, but have the Bernstein and generalized Markov properties.

Define for a smooth compact hypersurface $M^n$ of ${\mathbf{R}}^{n+1}$ its crumpleness $\kappa \left(M^n\right)$ as the ratio ${diam}_{{\mathbf{R}}^{n+1}}(M^n)/r(M^n)$, where $r\left(M^n\right)$ is the distance from $M^n$ to its central set. (In other words, $r\left(M^n\right)$ is the maximal radius of an open non-selfintersecting tube around $M^n$ in ${\mathbf{R}}^{n+1}.)$We prove that any $n$-dimensional non-singular compact algebraic hypersurface of degree $d$ is rigidly isotopic to an algebraic hypersurface of degree $d$ and of crumpleness $\le exp\left(c\right(n\left)d^{\alpha \left(n\right)d^{n+1}}\right)$. Here $c\left(n\right)$, $\alpha \left(n\right)$ depend only on $n$, and rigid isotopy means an isotopy passing only through hypersurfaces of degree...

Nel presente lavoro si studiano le applicazioni polinomiali proprie $$\phi :{\mathbb{R}}^n\to {\mathbb{R}}^q.$$ In particolare si prova: 1) se $\phi :{\mathbb{R}}^n\to \mathbb{R}$ è un'applicazione polinomiale tale che ${\phi }^{-1}\left(y\right)$ è compatto per ogni $y\in \mathbb{R}$, allora $\phi$ è propria; 2) se $\phi :{\mathbb{R}}^n\to {\mathbb{R}}^q$ è polinomiale a fibra compatta e $\phi \left({\mathbb{R}}^n\right)$ è chiuso in ${\mathbb{R}}^q$ allora $\phi$ è propria; 3) l'insieme delle applicazioni polinomiali proprie di ${\mathbb{R}}^n$ in ${\mathbb{R}}^q$ è denso, nella topologia $C^{\mathrm{\infty }}$, nello spazio delle applicazioni $C^{\mathrm{\infty }}$ di ${\mathbb{R}}^n$ in ${\mathbb{R}}^q$.

Let V ⊂ ℝⁿ, n ≥ 2, be an unbounded algebraic set defined by a system of polynomial equations $h₁\left(x\right)=\cdots =h_r\left(x\right)=0$ and let f: ℝⁿ→ ℝ be a polynomial. It is known that if f is positive on V then ${f|}_V$ extends to a positive polynomial on the ambient space ℝⁿ, provided V is a variety. We give a constructive proof of this fact for an arbitrary algebraic set V. Precisely, if f is positive on V then there exists a polynomial $h\left(x\right)={\sum }_{i=1}^{r}h{²}_i\left(x\right){\sigma }_i\left(x\right)$, where ${\sigma }_i$ are sums of squares of polynomials of degree at most p, such that f(x) + h(x) > 0 for x...

On s’intéresse aux difféomorphismes birationnels des surfaces algébriques réelles qui possèdent une dynamique réelle simple et une dynamique complexe riche. On donne un exemple d’une telle transformation sur ${\mathbb{P}}^1\times {\mathbb{P}}^1$, mais on montre qu’une telle situation est exceptionnelle et impose des conditions fortes à la fois sur la topologie du lieu réel et sur la dynamique réelle.

We give another proof of the fact that any semialbraic curve admits a tangential Markov inequality. We establish this inequality on semialgebraic surfaces with finitely many singular points.

In 1889 A. Markov proved that for every polynomial p in one variable the inequality $\left|\right|p^{\text{'}}{\left|\right|}_{[-1,1]}{\le \left(degp\right)²\left|\right|p\left|\right|}_{[-1,1]}$ is true. Moreover, the exponent 2 in this inequality is the best possible one. A tangential Markov inequality is a generalization of the Markov inequality to tangential derivatives of certain sets in higher-dimensional Euclidean spaces. We give some motivational examples of sets that admit the tangential Markov inequality with the sharp exponent. The main theorems show that the results on certain arcs and surfaces,...