Related papers: Polynomial maps with nilpotent Jacobians in dimens…

Suppose that a Lie type algebra L over a field K admits a Frobenius group of automorphisms FH with cyclic kernel F of order n and complement H such that the fixed-point subalgebra of F is trivial and the fixed-point subalgebra of H is…

In this work we discuss whether the non-commuting graph of a finite group can determine its nilpotency. More precisely, Abdollahi, Akbari and Maimani conjectured that if $G$ and $H$ are finite groups with isomorphic non-commuting graphs and…

We give a full classification of 6-dimensional nilpotent Lie algebras over an arbitrary field, including fields that are not algebraically closed and fields of characteristic~2. To achieve the classification we use the action of the…

It is known that every 3-dimensional noetherian Calabi-Yau algebra generated in degree 1 is isomorphic to a Jacobian algebra of a superpotential. Recently, S. P. Smith and the first author classified all superpotentials whose Jacobian…

Let H: C^2 -> C^2 be the Henon mapping given by (x,y) --> (p(x) - ay,x). The key invariant subsets are K_+/-, the sets of points with bounded forward images, J_+/- = the boundary of K_+/-, J = the union of J_+ and J_-, and K = the union of…

We give necessary and sufficient conditions, in the form of matrix identities, for a polynomial f in C[X,Y] to be a component of a polynomial automorphism of C^2 and to be a component of a Keller polynomial mapping of C^2, respectively…

Let $K$ be a field of characteristic different from $2$, $\bar{K}$ its algebraic closure. Let $n \ge 3$ be an odd prime such that $2$ is a primitive root modulo $n$. Let $f(x)$ and $h(x)$ be degree $n$ polynomials with coefficients in $K$…

The normal form for an n-dimensional map with irreducible nilpotent linear part is determined using sl2-representation theory. We sketch by example how the reducible case can also be treated in an algorithmic manner. The construction (and…

We propose an algebraic geometry framework for the Kakeya problem. We conjecture that for any polynomials $f,g\in\F_{q_0}[x,y]$ and any $\F_q/\F_{q_0}$, the image of the map $\F_q^3\to\F_q^3$ given by $(s,x,y)\mapsto…

Fix any algebraic extension $\mathbb K$ of the field $\mathbb Q$ of rationals. In this article we study exponential sets $V\subset \mathbb R^n$. Such sets are described by the vanishing of so called exponential polynomials, i.e.,…

The complex moment sequence m(P) is assigned to a univalent polynomial P by the Cauchy transform of the P(D), where D is the unit disk. We establish the representation of the Jacobian det dm(P) in terms of roots of the derivative P'.…

It is proved that the Jacobian of a k-endomorphism of k[x_1,...,x_n] over a field k of characteristic zero taking every tame coordinate to a coordinate, must be a nonzero constant in k. It is also proved that the Jacobian of an…

Let K be a field of characteristic zero, f(x) be a polynomial with coefficients in K and without multiple roots. We consider the superelliptic curve C_{f,q} defined by y^q=f(x), where q=p^r is a power of a prime p. We determine the Hodge…

This paper develops a theory of polynomial maps from commutative semigroups to arbitrary groups and proves that it has desirable formal properties when the target group is locally nilpotent. We apply this theory to solve Waring's Problem…

We study the Jacobian conjecture for Keller maps $f:X_0:=\mathbf{A}^n\rightarrow Y_0:=\mathbf{A}^n$ in characteristic $0$ and attempt to prove it. We are quite aware of the fact that many people have tried to prove the Jacobian conjecture…

Choose a random linear operator on a vector space of finite cardinality N: then the probability that it is nilpotent is 1/N. This is a linear analogue of the fact that for a random self-map of a set of cardinality N, the probability that…

A graph $G$ on $n$ vertices of diameter $D$ is called $H$-palindromic if $\alpha(G,k) = \alpha(G,D-k)$ for all $k=0, 1, \dots, \left \lfloor{\frac{D}{2}}\right \rfloor$, where $\alpha(G,k)$ is the number of unordered pairs of vertices at…

Let $R$ be a ring satisfying a polynomial identity and let $\delta$ be a derivation of $R$. We show that if $N$ is the nil radical of $R$ then $\delta(N)\subseteq N$ and the Jacobson radical of $R[x;\delta]$ is equal to $N[x;\delta]$. As a…

The theory of polynomial-like maps is of fundamental importance in holomorphic dynamics. We study dynamical properties of a larger class of maps. Our main result is that, under some natural conditions, a map of this class has a completely…

We describe the set of all locally nilpotent derivations of the quotient ring $\mathbb{K}[X,Y,Z]/(f(X)Y - \varphi(X,Z))$ constructed from the defining equation $f(X)Y = \varphi(X,Z)$ of a generalized Danielewski surface in $\mathbb K^3$ for…