Related papers: Polynomial maps with nilpotent Jacobians in dimens…

We give a geometric classification of $n$-dimensional nilpotent, commutative nilpotent and anticommutative nilpotent algebras. We prove that the corresponding geometric varieties are irreducible, find their dimensions and describe explicit…

Let $k$ be a field of any characteristic, $V$ a finite-dimensional vector space over $k$, and $S^d(V^*)$ be the $d$-th symmetric power of the dual space $V^*$. Given a linear map $\varphi$ on $V$ and an eigenvector $w$ of $\varphi$, we…

Let $K$ denote an algebraically closed field with characteristic 0, and let $q$ denote a nonzero scalar in $K$ that is not a root of unity. Let $A_q$ denote the unital associative $K$-algebra defined by generators $x,y$ and relations…

The present paper is devoted to investigating the two-dimensional real Jacobian conjecture. This conjecture claims that if $F=\left(f,g\right):\mathbb{R}^2\rightarrow \mathbb{R}^2$ is a polynomial map with $\det DF\left(x,y\right)\ne0$ for…

Let $P_n=k[x_1,x_2,\ldots,x_n]$ be the polynomial algebra over a field $k$ of characteristic zero in the variables $x_1,x_2,\ldots,x_n$ and $\mathscr{L}_n$ be the left-symmetric algebra of all derivations of $P_n$ \cite{Dzhuma99,UU2014-1}.…

Let $k$ be an algebraically closed field of characteristic $p >0$. We consider the variety of nilpotent pairs $(A,B)$ with $[A,B]=\lambda I$, namely the set of pairs $ X = \{ (A,B) \in M_n(k) \times M_n(k) \mid A,B \text{ nilpotent},…

Let K[x,y] be the algebra of two-variable polynomials over a field K. A polynomial p=p(x, y) is called a test polynomial (for automorphisms) if, whenever \phi(p)=p for a mapping \phi of K[x,y], this \phi must be an automorphism. Here we…

Let U(L) be the enveloping algebra of a finite dimensional Lie algebra L over a field k of characteristic zero, Z(U(L)) its center and Sz(U(L)) its semicenter. A sufficient condition is given in order for Sz(U(L)) to be a polynomial algebra…

In this paper we prove the generalized Kaplansky conjecture for the Jordan algebras of the type $J_n$ in particular for self adjoint $2\times 2$ matrices over $\R$, over $\C$, $\HH$ and $\Oct$. In fact, we prove that the image of…

Let $\mathbb K$ be an algebraically closed field of characteristic zero. Let $V$ be a module over the polynomial ring $\mathbb K[x,y]$. The actions of $x$ and $y$ determine linear operators $P$ and $Q$ on $V$ as a vector space over $\mathbb…

Let $n\geq 2$ and $\mathbb K $ be a number field of characteristic $0$. Jacobian Conjecture asserts for a polynomial map $\mathcal P$ from $\mathbb K ^n$ to itself, if the determinant of its Jacobian matrix is a nonzero constant in $\mathbb…

We construct some examples of polynomial maps over finite fields that admit subvarieties with a peculiar property: every geometric point is mapped to a fixed point by some iteration of the map, while the whole subvariety is not. Several…

Let (K, v) be a henselian valued field of arbitrary rank. In this paper, we give an irreducibility criterion for multivariate polynomials over K using valuation theory.

This paper deals with the variety of commutative nonassociative algebras satisfying the identity $L_x^3+ \gamma L_{x^3} = 0$, $\gamma \in K$. Correa et al proved that if $\gamma = 0,1$ then any such finitely generated algebra is nilpotent.…

In this article, we classify invariants and conjugacy classes of triangular polynomial maps. We make these classifications in dimension 2 over domains containing $\Q$, dimension 2 over fields of characteristic $p$, and dimension 3 over…

A subspace $X$ of a vector space over a field $K$ is hyperinvariant with respect to an endomorphism $f$ of $V$ if it is invariant for all endomorphisms of $V$ that commute with $f$. We assume that $f$ is locally nilpotent, that is, every $…

For a polynomial map $\mathbf{f} : k^n \to k^m$ ($k$ a field), we investigate those polynomials $g \in k[t_1,\ldots, t_n]$ that can be written as a composition $g = h \circ \mathbf{f}$, where $h: k^m \to k$ is an arbitrary function. In the…

The Jacobian Conjecture uses the equation $det(Jac(F))\in k^*$, which is a very short way to write down many equations putting restrictions on the coefficients of a polynomial map $F$. In characteristic $p$ these equations do not suffice to…

We show that the iterated images of a Jacobian pair stabilize; that is, the k-th iterates of a polynomial map of complex two-space to itself with a nonzero constant Jacobian determinant all have the same image for sufficiently large k. More…

We consider pairs (V,H) of subgroups of a connected unipotent complex Lie group G for which the induced VxH-action on G by multiplication from the left and from the right is free. We prove that this action is proper if the Lie algebra g of…