Let X, Y be complex affine varieties and f:X → Y a regular mapping. We prove that if dim X ≥ 2 and f is closed in the Zariski topology then f is proper in the classical topology.

We present a class of counterexamples to the Cancellation Problem over arbitrary commutative rings, using non-free stably free modules and locally nilpotent derivations.

2000 Mathematics Subject Classification: Primary: 14R10. Secondary: 14R20, 13N15.Let R be a UFD containing a field of characteristic 0, and Bm = R[Y1, . . . , Ym] be a polynomial ring over R. It was conjectured in [5] that if D is an R-elementary monomial derivation of B3 such that ker D is a finitely generated R-algebra then the generators of ker D can be chosen to be linear in the Yi ’s. In this paper, we prove that this does not hold for B4. We also investigate R-elementary derivations D of Bm...

A. Crachiola and L. Makar-Limanov [J. Algebra 284 (2005)] showed the following: if X is an affine curve which is not isomorphic to the affine line ${¹}_k$, then ML(X×Y) = k[X]⊗ ML(Y) for every affine variety Y, where k is an algebraically closed field. In this note we give a simple geometric proof of a more general fact that this property holds for every affine variety X whose set of regular points is not k-uniruled.

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 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.

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 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...

We construct and study length 2 variables of A[x,y] (A is a commutative ring). If A is an integral domain, we determine among these variables those which are tame. If A is a UFD, we prove that these variables are all stably tame. We apply this construction to show that some polynomials of A[x₁,...,xₙ] are variables using transfer.

We show that if X, Y are smooth, compact k-dimensional submanifolds of ℝⁿ and 2k+2 ≤ n, then each diffeomorphism ϕ: X → Y can be extended to a diffeomorphism Φ: ℝⁿ → ℝⁿ which is tame (to be defined in this paper). Moreover, if X, Y are real analytic manifolds and the mapping ϕ is analytic, then we can choose Φ to be also analytic. We extend this result to some interesting categories of closed (not necessarily compact) subsets of ℝⁿ, namely, to the category of Nash submanifolds...

We present an example which confirms the assertion of the title.