Let V, W be algebraic subsets of $k^n$, $k^m$ respectively, with n ≤ m. It is known that any finite polynomial mapping f: V → W can be extended to a finite polynomial mapping $F:k^n\to k^m.$ The main goal of this paper is to estimate from above the geometric degree of a finite extension $F:k^n\to k^n$ of a dominating mapping f: V → W, where V and W are smooth algebraic sets.

Introduite par Witt en 1937, la théorie des formes quadratiques sur un corps joue un rôle central dans la démonstration des conjectures de Milnor par Voevodsky via les travaux pionniers de Rost qui y interviennent. Réciproquement, les méthodes de Rost et Voevodsky utilisant la théorie des motifs et les opérations de Steenrod motiviques révolutionnent la théorie des formes quadratiques et ont conduit à la démonstration de résultats de base qui semblaient auparavant inaccessibles. On expliquera notamment...

We recall first Mather's Lemma providing effective necessary and sufficient conditions for a connected submanifold to be contained in an orbit. We show that two homogeneous polynomials having isomorphic Milnor algebras are right-equivalent.

We prove that every injective endomorphism of an affine algebraic variety over an algebraically closed field of characteristic zero is an automorphism. We also construct an analytic curve in ℂ⁶ and its holomorphic bijection which is not a biholomorphism.

A broad family of involutive birational transformations of an open dense subset of ${ℝ}^n$ onto itself is constructed explicitly. Examples with arbitrarily high complexity are presented. Construction of birational transformations such that ${\phi }^k=\mathrm{Id}$ for a fixed integer $k>2$ is also presented.

Linear fractional recurrences are given as $z_{n+k}=A\left(z\right)/B\left(z\right)$, where $A\left(z\right)$ and $B\left(z\right)$ are linear functions of $z_n,z_{n+1},\cdots ,z_{n+k-1}$. In this article we consider two questions about these recurrences: (1) Find $A\left(z\right)$ and $B\left(z\right)$ such that the recurrence is periodic; and (2) Find (invariant) integrals in case the induced birational map has quadratic degree growth. We approach these questions by considering the induced birational map and determining its dynamical degree. The first theorem shows that for each $k$ there are $k$-step linear fractional recurrences...

We show that every n-dimensional smooth algebraic variety X can be covered by Zariski open subsets $U_i$ which are isomorphic to closed smooth hypersurfaces in ${ℂ}^{n+1}$. As an application we show that forevery (pure) n-1-dimensional ℂ-uniruled variety $X\subset {ℂ}^m$ there is a generically-finite (even quasi-finite) polynomial mapping $f:{ℂ}^n\to {ℂ}^m$ such that $X\subset S_f$. This gives (together with [3]) a full characterization of irreducible components of the set $S_f$ for generically-finite polynomial mappings $f:{ℂ}^n\to {ℂ}^m$.

Suppose that $R\subset S$ are regular local rings which are essentially of finite type over a field $k$ of characteristic zero. If $V$ is a valuation ring of the quotient field $K$ of $S$ which dominates $S$, then we show that there are sequences of monoidal transforms (blow ups of regular primes) $R\to R_1$ and $S\to S_1$ along $V$ such that $R_1\to S_1$ is a monomial mapping. It follows that a morphism of nonsingular varieties can be made to be a monomial mapping along a valuation, after blow ups of nonsingular subvarieties.