Related papers: On pairs of commuting derivations of the polynomia…

Let $A$ be the polynomial ring over $k$ (a field of characteristic zero) in $n+1$ variables. The commuting derivations conjecture states that $n$ commuting locally nilpotent derivations on $A$, linearly independent over $A$, must satisfy…

We study a monomial derivation $d$ proposed by J. Moulin Ollagnier and A. Nowicki in the polynomial ring of four variables, and prove that $d$ has no Darboux polynomials if and only if $d$ has a trivial field of constants.

In this paper we show that the image of any locally finite $k$-derivation of the polynomial algebra $k[x, y]$ in two variables over a field $k$ of characteristic zero is a Mathieu subspace. We also show that the two-dimensional Jacobian…

We study the problem of characterizing polynomial vector fields that commute with a given polynomial vector field on a plane. It is a classical result that one can write down solution formulas for an ODE that corresponds to a planar vector…

The first part of the paper will describe a recent result of K. Retert in (\cite{Ret}) for $k[x_1,\ldots,x_n]$ and $k[[x_1,\ldots,x_n]]$. This result states that if $\mathfrak{D}$ is a set of commute $k$-derivations of $k[x,y]$ such that…

We first study some properties of images of commuting differential operators of polynomial algebras of order one with constant leading coefficients. We then propose what we call the image conjecture on these differential operators and show…

We study the subgroup of $k$-automorphisms of $k[x,y]$ which commute with a simple derivation $D$ of $k[x,y].$ We prove, for example, that this subgroup is trivial when $D$ is a Shamsuddin simple derivation. In the general case of simple…

We consider differential rings of the form (K[x; y];D), where K is an algebraically closed field of characteristic zero and D : K[x; y] \to K[x; y] is a K-derivation. We study the Automorphism Group of such a ring and give criteria for…

In this article we study the field of rational constants and Darboux polynomials of a generalized cyclotomic $K$-derivation $d$ of $K[X]$. It is shown that $d$ is without Darboux polynomials if and only if $K(X)^d=K$. Result is also studied…

Let $k[X]=k[x_0,...,x_{n-1}]$ and $k[Y]=k[y_0,...,y_{n-1}]$ be the polynomial rings in $n\geqslant 3$ variables over a field $k$ of characteristic zero containing the $n$-th roots of unity. Let $d$ be the cyclotomic derivation of $k[X]$,…

Polynomials in differentiation operators are considered. The Darboux transformations covariance determines non-Abelian entries to form the coefficients of the polynomials. Joint covariance of a pair of such polynomials (Lax pair) as a…

Let $\mathbb K$ be an algebraically closed field of characteristic zero, $\mathbb K[x, y]$ the polynonial ring in variables $x$, $y$ and let $W_2(\mathbb K)$ be the Lie algebra of all $\mathbb K$-derivations on $\mathbb K[x, y]$. A…

In the paper, we first study the subgroup of $ K$-automorphisms of $K[x_1,\allowbreak \ldots,x_n]$ which commutes with a simple derivation of $K[x_1,\ldots,x_n]$. We show that the subgroup of $ K$-automorphisms of $K[x_1,\ldots,x_n]$ which…

Let K be a field of characteristic zero. We prove that images of a linear K-derivation and a linear K-E-derivation of the ring K[x 1 ,x 2 ,x 3 ] of polynomial in three variables over K are Mathieu-Zhao subspaces, which affirms the LFED…

Let $\mathbb{F}$ be a finite field and let $f$ be a linear polynomial in $\mathbb{F}[x]$. We investigate the number of polynomials of degree $d$ which commute with $f$ under composition. In so doing, we rediscover a result of Park, but with…

Let $k$ be a field of characteristic zero. Let $m$ and $\alpha$ be positive integers. For $n\geq 2$, let $R_n=k[x_1,x_2,\dots,x_n]$ with the $k$-derivation $d_n$ given by…

Miyanishi proved that the ring of invariants of any $\mathbb{G}_a$ action on $\mathbb{A}^3$ is $\mathbb{A}^2$, when the field $k$ has zero characteristic. However, it is not known if this result holds when $k$ has positive characteristic.…

$ $Let $k$ be a field of characteristic zero. If $c_1, c_2\in k\setminus \{0\}, s,t\geq 1$ and $u\geq 0$, then it is shown that the $k$-derivations $\partial_x + x^u(c_1x^ty^s+c_2)\partial_y$ and $\partial_x +…

Let $ K[x, y]$ be the polynomial algebra in two variables over a field $K$ of characteristic $0$. A subalgebra $R$ of $K[x, y]$ is called a retract if there is an idempotent homomorphism (a {\it retraction}, or {\it projection}) $\varphi:…

Let K be an algebraically closed field of characteristic zero and let f(x,y) be a nonzero polynomial of K[x,y]. We prove that if the generic element of the family $(f-\lambda)\_{\lambda}$ is a rational polynomial, and if the Jacobian J(f,g)…