Let K be a unique factorization domain of characteristic p > 0, and let f ∈ K[x₁,...,xₙ] be a polynomial not lying in $K[x{₁}^p,...,x{ₙ}^p]$. We prove that $K[x{₁}^p,...,x{ₙ}^p,f]$ is the ring of constants of a K-derivation of K[x₁,...,xₙ] if and only if all the partial derivatives of f are relatively prime. The proof is based on a generalization of Freudenburg’s lemma to the case of polynomials over a unique factorization domain of arbitrary characteristic.

We obtain two equivalent conditions for m polynomials in n variables to form a p-basis of a ring of constants of some polynomial K-derivation, where K is a unique factorization domain of characteristic p > 0. One of these conditions involves Jacobians while the other some properties of factors. In the case m = n this extends the known theorem of Nousiainen, and we obtain a new formulation of the Jacobian conjecture in positive characteristic.

Sullivan associated a uniquely determined $DGA{|}_{\mathbf{Q}}$ to any simply connected simplicial complex $E$. This algebra (called minimal model) contains the total (and exactly) rational homotopy information of the space $E$. In case $E$ is the total space of a principal $G$-bundle, ($G$ is a compact connected Lie-group) we associate a $G$-equivariant model $U_G\left[E\right]$, which is a collection of “$G$-homotopic” $DGA$’s${|}_{\mathbf{R}}$ with $G$-action. $U_G\left[E\right]$ will, in general, be different from the Sullivan’s minimal model of the space $E$. $U_G\left[E\right]$ contains the total rational...

2000 Mathematics Subject Classification: 13N15, 13A50, 16W25.We reduce the Nowicki conjecture on Weitzenböck derivations of polynomial algebras to a well known problem of classical invariant theory.

Let A be a commutative algebra without zero divisors over a field k. If A is finitely generated over k, then there exist well known characterizations of all k-subalgebras of A which are rings of constants with respect to k-derivations of A. We show that these characterizations are not valid in the case when the algebra A is not finitely generated over k.

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...

If $X$ is a smooth scheme over a perfect field of characteristic $p$, and if ${𝒟}_X^{\left(\infty \right)}$ is the sheaf of differential operators on $X$ [7], it is well known that giving an action of ${𝒟}_X^{\left(\infty \right)}$ on an ${𝒪}_X$-module $ℰ$ is equivalent to giving an infinite sequence of ${𝒪}_X$-modules descending $ℰ$ via the iterates of the Frobenius endomorphism of $X$ [5]. We show that this result can be generalized to any infinitesimal deformation $f:X\to S$ of a smooth morphism in characteristic $p$, endowed with Frobenius liftings. We also show that it extends to adic...

At first we prove some results on a general polynomial derivation using few results of linear derivation. Then we study the ring of constants of a linear derivation for some rings. We know that any linear derivation is a nonsimple derivation. In the last section we find the smallest integer $w>1$ such that the polynomial ring in $n$ variables is $w$-differentially simple, all $w$ derivations are nonsimple and the $w$ derivations set contains a linear derivation.

Free Poisson algebras are very closely connected with polynomial algebras, and the Poisson brackets are used to solve many problems in affine algebraic geometry. In this note, we study Poisson derivations on the symplectic Poisson algebra, and give a connection between the Jacobian conjecture with derivations on the symplectic Poisson algebra.

We observe that the characterization of rings of constants of derivations in characteristic zero as algebraically closed subrings also holds in positive characteristic after some natural adaptation. We also present a characterization of such rings in terms of maximality in some families of rings.

A concept of a slice of a semisimple derivation is introduced. Moreover, it is shown that a semisimple derivation d of a finitely generated commutative algebra A over an algebraically closed field of characteristic 0 is nothing other than an algebraic action of a torus on Max(A), and, using this, that in some cases the derivation d is linearizable or admits a maximal invariant ideal.

We connect the theorems of Rentschler [rR68] and Dixmier [jD68] on locally nilpotent derivations and automorphisms of the polynomial ring $A_0$ and of the Weyl algebra $A_1$, both over a field of characteristic zero, by establishing the same type of results for the family of algebras $$A_h=\langle x,y\mid yx-xy=h\left(x\right)\rangle ,$$ where $h$ is an arbitrary polynomial in $x$. In the second part of the paper we consider a field $𝔽$ of prime characteristic and study $𝔽\left[t\right]$comodule algebra structures on $A_h$. We also compute the Makar-Limanov invariant of absolute constants...

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.)