Narrow Results

Let $R$ be a commutative integral domain with unity that contains $\mathbb{Q}$. In this paper we describe the $R$-automorphism group of $R$-algebras of the form $R[x,y,z]/(c(x)z-q(x,y))$, where $c(x)\in R[x]$ is quasi-monic and $q(x,y)\in R[x,y]$ is quasi-monic of degree at least two with respect to $y$.

An affine algebraic variety $X$ is rigid if the algebra of regular functions $𝕂[X]$ admits no nonzero locally nilpotent derivation. We prove that a factorial trinomial hypersurface is rigid if and only if every exponent in the trinomial is at least $2$.

By an additive action on an algebraic variety $X$ we mean a regular effective action ${𝔾}_a^n\times X\to X$ with an open orbit of the commutative unipotent group ${𝔾}_a^n$. In this paper, we give a classification of additive actions on complete toric surfaces.

Let $A$ be a reduced ring containing $\mathbb{Q}$ and let ${\xi }_1,{\xi }_2$ be commuting locally nilpotent derivations of $A$. In this paper, we give an algorithm to decide the local nilpotency of derivations of the form $\xi =a_1{\xi }_1+a_2{\xi }_2$, where $a_1,a_2$ are elements in $A$.

We study commutative associative polynomial operations ${𝔸}^n\times {𝔸}^n\to {𝔸}^n$ with unit on the affine space ${𝔸}^n$ over an algebraically closed field of characteristic zero. A classification of such operations is obtained up to dimension 3. Several series of operations are constructed in arbitrary dimension. Also we explore a connection between commutative algebraic monoids on affine spaces and additive actions on toric varieties.

In this paper, we contrast the structure of a noncommutative algebra R with that of the skew power series ring R[[y;d]]. Several of our main results examine when the rings R, R^d, and R[[y;d]] are prime or semiprime under the assumption that d is a locally nilpotent derivation.

A short proof is given of Rudakov’s result (announced in [A. N. Rudakov, Subalgebras and automorphisms of Lie algebras of Cartan type, Funktsional. Anal. i Prilozhen. 20(1) (1986) 83–84]), that the group of automorphisms of the Lie algebra ${\mathrm{\text{Der }}}_K(P_n)$ of derivations of a polynomial algebra $P_n=K[x_1,\dots ,x_n]$, over a field of characteristic zero is canonically isomorphic to the group of automorphisms of the polynomial algebra $P_n$.

Let $R\subset A$ be factorial domains containing $\mathbb{Q}$. In this paper, we give a criterion, in terms of locally nilpotent derivations, for $A$ to be $R$-isomorphic to $R[v,w]/(cw-h(v))$, where $c\in R$ is nonzero and $h(v)\in R[v]$. As a consequence, we retrieve a recent result due to Masuda [Families of hypersurfaces with noncancellation property, Proc. Amer. Math. Soc.145(4) (2017) 1439–1452] characterizing Danielewski hypersurfaces whose coordinate ring is factorial. We also apply our criterion to the study of triangularizable locally nilpotent $R$-derivations of the polynomial ring in two variables over $R$.