It is explained that the following two problems are equivalent: (i) describing all affine rulings of any given weighted projective plane; (ii) describing all weighted-homogeneous locally nilpotent derivations of k[X,Y,Z]. Then the solution of (i) is sketched. (Outline of our joint work with Peter Russell.)

We classify all finitely generated integral algebras with a rational action of a reductive group such that any invariant subalgebra is finitely generated. Some results on affine embeddings of homogeneous spaces are also given.

We present an example of finite mappings of algebraic varieties f:V → W, where V ⊂ kⁿ, $W\subset k^{n+1}$, and $F:kⁿ\to k^{n+1}$ such that ${F|}_V=f$ and gdeg F = 1 < gdeg f (gdeg h means the number of points in the generic fiber of h). Thus, in some sense, the result of this note improves our result in J. Pure Appl. Algebra 148 (2000) where it was shown that this phenomenon can occur when V ⊂ kⁿ, $W\subset k^m$ with m ≥ n+2. In the case V,W ⊂ kⁿ a similar example does not exist.

A relationship between jacobian maps and the commutativity properties of suitable couples of hamiltonian vector fields is studied. A theorem by Meisters and Olech is extended to the nonpolynomial case. A property implying the Jacobian Conjecture in ℝ² is described.

We develop a new combinatorial method to deal with a degree estimate for subalgebras generated by two elements in different environments. We obtain a lower bound for the degree of the elements in two-generated subalgebras of a free associative algebra over a field of zero characteristic. We also reproduce a somewhat refined degree estimate of Shestakov and Umirbaev for the polynomial algebra, which plays an essential role in the recent celebrated solution of the Nagata conjecture and the strong...

We consider the family of polynomials in $\mathbf{C}[x,y,z]$ of the form $x^{2}y-z^2-xq(x,z)$. Two such polynomials $P_1$ and $P_2$ are equivalent if there is an automorphism ${\varphi }^{*}$ of $\mathbf{C}[x,y,z]$ such that ${\varphi }^{*}\left(P_1\right)=P_2$. We give a complete classification of the equivalence classes of these polynomials in the algebraic and analytic category. As a consequence, we find the following results. There are explicit examples of inequivalent polynomials $P_1$ and $P_2$ such that the zero set of $P_1+c$ is isomorphic to the zero set of $P_2+c$ for all $c\in \mathbf{C}$. There exist polynomials which are algebraically...

We consider singular $\mathbb{Q}$-acyclic surfaces with smooth locus of non-general type. We prove that if the singularities are topologically rational then the smooth locus is ${ℂ}^1$- or ${ℂ}^{*}$-ruled or the surface is up to isomorphism one of two exceptional surfaces of Kodaira dimension zero. For both exceptional surfaces the Kodaira dimension of the smooth locus is zero and the singular locus consists of a unique point of type $A_1$ and $A_2$ respectively.

Soit $k$ un corps de caractéristique nulle, $P$ un polynôme de Laurent en $d$ variables, à coefficients dans $k$ et non dégénéré pour son polyèdre de Newton à l’infini. Soit $d$ fonctions non constantes $f_l$ à variables séparées et définies sur des variétés lisses. A la manière de Guibert, Loeser et Merle, dans le cas local, nous calculons dans cet article, la fibre de Milnor motivique à l’infini de la composée $P\left(f\right)$ en termes du polyèdre de Newton à l’infini de $P$. Pour $P$ égal à la somme $x_1+x_2$ nous obtenons une formule...

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.

We study the LND conjecture concerning the images of locally nilpotent derivations, which arose from the Jacobian conjecture. Let $R$ be a domain containing a field of characteristic zero. We prove that, when $R$ is a one-dimensional unique factorization domain, the image of any locally nilpotent $R$-derivation of the bivariate polynomial algebra $R[x,y]$ is a Mathieu-Zhao subspace. Moreover, we prove that, when $R$ is a Dedekind domain, the image of a locally nilpotent $R$-derivation of $R[x,y]$ with some additional conditions...

A polynomial map F = (P,Q) ∈ ℤ[x,y]² with Jacobian $JF:=P_x{Q}_y-P_y{Q}_x\equiv 1$ has a polynomial inverse with integer coefficients if the complex plane curve P = 0 has infinitely many integer points.

We prove a sufficient condition for the Jacobian problem in the setting of real, complex and mixed polynomial mappings. This follows from the study of the bifurcation locus of a mapping subject to a new Newton non-degeneracy condition.

Let $G\to GL\left(V\right)$ be a representation of a reductive linear algebraic group $G$ on a finite-dimensional vector space $V$, defined over an algebraically closed field of characteristic zero. The categorical quotient $X=V//G$ carries a natural stratification, due to D. Luna. This paper addresses the following questions:(i) Is the Luna stratification of $X$ intrinsic? That is, does every automorphism of $V//G$ map each stratum to another stratum?(ii) Are the individual Luna strata in $X$ intrinsic? That is, does every automorphism...