Related papers: A degree bound for strongly nilpotent polynomial a…

Let $X, Y$ be smooth algebraic varieties of the same dimension. Let $f, g : X \to Y$ be finite polynomial mappings. We say that $f, g$ are equivalent if there exists a regular automorphism $\Phi \in Aut(X)$ such that $f = g\circ \Phi$. Of…

Let $F$ be a field, char$(F)\neq 2$. Then every finite-dimensional $F$-algebra has either an idempotent or an absolute nilpotent if and only if over $F$ every polynomial of odd degree has a root in $F$. This is also necessary and sufficient…

A Lie algebra L is known to be nilpotent if it admits a grading by (Zp, +) with support X not containing 0. It is also known that the class of L can be bounded by some explicit function of |X|. We generalise this and other classical results…

Let alpha be an automorphism of a hyperelliptic curve C of genus g, and let alpha' be the automorphism of P^1 induced by alpha. Let n be the order of alpha and let n' be the order of alpha'. We show that the triple (g,n,n') completely…

In this paper, we investigate the existence of fixed-point-free automorphisms for finite-dimensional Lie algebras. By a result of Jacobson, a Lie algebra admitting a fixed-point-free automorphism is solvable. We prove that such a Lie…

This paper is devoted to the study of graded associative algebras that satisfy a graded polynomial identity of degree $2$. % Let $\mathsf{G}$ be a finite abelian group, $\mathbb{F}$ a field of characteristic zero and $\mathfrak{A}$ a…

We prove nilpotency results for Lie algebras over an arbitrary field admitting a derivation, which satisfies a given polynomial identity $r(t)=0$. For the polynomial $r=t^n-1$ we obtain results on the nilpotency of Lie algebras admitting a…

We call a group $G$ nilpotently Jordan of class at most $c$ $(c\in\mathbb{N})$ if there exists a constant $J\in\mathbb{Z}^+$ such that every finite subgroup $H\leqq G$ contains a nilpotent subgroup $K\leqq H$ of class at most $c$ and index…

In this article we extend results of Zomorrodian to determine upper bounds for the order of a nilpotent group of automorphisms of a complex $d$-dimensional family of compact Riemann surfaces, where $d \geqslant 1.$ We provide conditions…

Let $I$ be the ideal of relations between the leading terms of the polynomials defining an automorphism of $K^n$. In this paper, we prove the existence of a locally nilpotent derivation which preserves $I$. Moreover, if $I$ is principal,…

An important invariant of a polynomial $f$ is its Jacobian algebra defined by its partial derivatives. Let $f$ be invariant with respect to the action of a finite group of diagonal symmetries $G$. We axiomatically define an orbifold…

Let $K$ be an {\em arbitrary} field of characteristic $p>0$, let $A$ be one of the following algebras: $P_n:= K[x_1, ..., x_n]$ is a polynomial algebra, $\CD (P_n)$ is the ring of differential operators on $P_n$, $\CD (P_n)\t P_m$, the…

Consider the gradient map associated to any non-constant homogeneous polynomial $f\in \C[x_0,...,x_n]$ of degree $d$, defined by \[\phi_f=grad(f): D(f)\to \CP^n, (x_0:...:x_n)\to (f_0(x):...:f_n(x))\] where $D(f)=\{x\in \CP^n; f(x)\neq 0\}$…

We construct a non-proper set of two variables polynomial maps and study the nowhere vanishing Jacobian condition of the Jacobian conjecture for this set. We obtain some classes of polynomial maps satisfying the 2-dimensional Jacobian…

In this article we study maps with nilpotent Jacobian in $\mathbb{R}^n$ distinguishing the cases when the rows of $JH$ are linearly dependent over $\mathbb{R}$ and when they are linearly independent over $\mathbb{R}.$ In the linearly…

Let $K$ be any field with $\textup{char}K\neq 2,3$. We classify all cubic homogeneous polynomial maps $H$ over $K$ with $\textup{rk} JH\leq 2$. In particular, we show that, for such an $H$, if $F=x+H$ is a Keller map then $F$ is invertible,…

Let V be a variety of not necessarily associative algebras, and A an inverse limit of nilpotent algebras A_i\in V, such that some finitely generated subalgebra S \subseteq A is dense in A under the inverse limit of the discrete topologies…

Suppose that a metacyclic Frobenius group $FH$, with kernel $F$ and complement $H$, acts by automorphisms on a finite group $G$, in such a way that $C_G(F)$ is trivial and $C_G(H)$ is nilpotent. It is known that $G$ is nilpotent and its…

Let $f$ be a polynomial automorphism of the affine plane. In this paper we consider the possibility for it to possess infinitely many periodic points on an algebraic curve $C$. We conjecture that this happens if and only if $f$ admits a…

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…