ON THE CONNECTION BETWEEN DIFFERENTIAL POLYNOMIAL RINGS AND POLYNOMIAL RINGS OVER NIL RINGS

In this paper, we study some connections between the polynomial ring $R[y]$ and the differential polynomial ring $R[x;D]$. In particular, we answer a question posed by Smoktunowicz, which asks whether $R[y]$ is nil when $R[x;D]$ is nil, provided that $R$ is an algebra over a field of positive characteristic and $D$ is a locally nilpotent derivation.

Pascal finite polynomial automorphisms

The class of Pascal finite polynomial automorphisms is a subclass of the class of locally finite ones allowing a more effective approach. In characteristic zero, a Pascal finite automorphism is the exponential map of a locally nilpotent derivation. However, Pascal finite automorphisms are defined in any characteristic, and therefore constitute a generalization of exponential automorphisms to positive characteristic. In this paper, we prove several properties of Pascal finite automorphisms. We obtain in particular that the Pascal finite property is stable under taking powers but not under composition. This leads us to formulate a generalization of the exponential generators conjecture to arbitrary characteristic.

THE FACTORIAL CONJECTURE AND IMAGES OF LOCALLY NILPOTENT DERIVATIONS

The factorial conjecture was proposed by van den Essen et al. [‘On the image conjecture’, J. Algebra 340(1) (2011), 211–224] to study the image conjecture, which arose from the Jacobian conjecture. We show that the factorial conjecture holds for all homogeneous polynomials in two variables. We also give a variation of the result and use it to show that the image of any linear locally nilpotent derivation of $\mathbb{C}[x,y,z]$ is a Mathieu–Zhao subspace.

ON SEPARABLE AND -FORMS

In this paper, we will prove that any $\mathbb{A}^{3}$-form over a field $k$ of characteristic zero is trivial provided it has a locally nilpotent derivation satisfying certain properties. We will also show that the result of Kambayashi on the triviality of separable $\mathbb{A}^{2}$-forms over a field $k$ extends to $\mathbb{A}^{2}$-forms over any one-dimensional Noetherian domain containing $\mathbb{Q}$.

On rigidity of factorial trinomial hypersurfaces

An affine algebraic variety [Formula: see text] is rigid if the algebra of regular functions [Formula: see text] 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 [Formula: see text].

SKEW POWER SERIES RINGS OF DERIVATION TYPE

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, Rd, and R[[y;d]] are prime or semiprime under the assumption that d is a locally nilpotent derivation.

THE CONJECTURE OF NOWICKI ON WEITZENBÖCK DERIVATIONS OF POLYNOMIAL ALGEBRAS

The Weitzenböck theorem states that if Δ is a linear locally nilpotent derivation of the polynomial algebra K[Z] = K[z1,…,zm] over a field K of characteristic 0, then the algebra of constants of Δ is finitely generated. If m = 2n and the Jordan normal form of Δ consists of 2 × 2 Jordan cells only, we may assume that K[Z] = K[X,Y] and Δ(yi) = xi, Δ(xi) = 0, i = 1,…,n. Nowicki conjectured that the algebra of constants K[X,Y]Δ is generated by x1,…,xn and xiyj – xjyi, 1 ≤ i < j ≤ n. Recently this conjecture was confirmed in the Ph.D. thesis of Khoury with a very computational proof, and also by Derksen whose proof is based on classical results of invariant theory. In this paper we give an elementary proof of the conjecture of Nowicki which does not use any invariant theory. Then we find a very simple system of defining relations of the algebra K[X,Y]Δ which corresponds to the reduced Gröbner basis of the related ideal with respect to a suitable admissible order, and present an explicit basis of K[X,Y]Δ as a vector space.