Let d₃ ≥ p₂ > p₁ ≥ 3 be integers such that p₁,p₂ are prime numbers. We show that the sequence (p₁,p₂,d₃) is the multidegree of some tame automorphism of ℂ³ if and only if d₃ ∈ p₁ℕ + p₂ℕ, i.e. if and only if d₃ is a linear combination of p₁ and p₂ with coefficients in ℕ.

In this article, we prove that a $\mathbb{Q}$-homology plane $X$ with two algebraically independent $G_a$-actions is isomorphic to either the affine plane or a quotient of an affine hypersurface $xy=z^m-1$ in the affine $3$-space via a free $\mathbb{Z}/m\mathbb{Z}$-action, where $m$ is the order of a finite group $H_1(X;\mathbb{Z})$.

We describe the polynomials P ∈ ℂ[x,y] such that $P(1/vⁿ,A₁vⁿ+A₂v^{2n}+...+A_{m-1}{v}^{n(m-1)}+v^{nm-k}w)\in ℂ[v,w]$. As applications we give new examples of bad field generators and examples of families of polynomials with smooth and irreducible fibers.

We give a description of the set of points for which the Fedoryuk condition fails in terms of the Łojasiewicz exponent at infinity near a fibre of a polynomial.

Let k be an algebraically closed field of characteristic zero and $F:=x+{\left(Ax\right)}^{*d}:kⁿ\to kⁿ$ a Drużkowski mapping of degree ≥ 2 with det JF = 1. We classify all such mappings whose Jacobian matrix JF is symmetric. It follows that the Jacobian Conjecture holds for these mappings.

Let 𝕂 denote ℝ or ℂ, n > 1. The Jacobian Conjecture can be formulated as follows: If F:𝕂ⁿ → 𝕂ⁿ is a polynomial map with a constant nonzero jacobian, then F is a polynomial automorphism. Although the Jacobian Conjecture is still unsolved even in the case n = 2, it is convenient to consider the so-called Generalized Jacobian Conjecture (for short (GJC)): the Jacobian Conjecture holds for every n>1. We present the reduction of (GJC) to the case of F of degree 3 and of symmetric homogeneous...

There are nontrivial dualities and parallels between polynomial algebras and the Grassmann algebras (e.g., the Grassmann algebras are dual of polynomial algebras as quadratic algebras). This paper is an attempt to look at the Grassmann algebras at the angle of the Jacobian conjecture for polynomial algebras (which is the question/conjecture about the $$Jacobian set– the set of all algebra endomorphisms of a polynomial algebra with the Jacobian $1$ – the Jacobian conjecture claims that the Jacobian...

We prove that the study of the Łojasiewicz exponent at infinity of overdetermined polynomial mappings $ℂⁿ\to {ℂ}^m$, m > n, can be reduced to the one when m = n.

Some estimates of the Łojasiewicz gradient exponent at infinity near any fibre of a polynomial in two variables are given. An important point in the proofs is a new Charzyński-Kozłowski-Smale estimate of critical values of a polynomial in one variable.

We show that every local polynomial diffeomorphism (f,g) of the real plane such that deg f ≤ 3, deg g ≤ 3 is a global diffeomorphism.

Assume that X,Y are integral noetherian affine schemes. Let f:X → Y be a dominant, generically finite morphism of finite type. We show that the set of points at which the morphism f is not finite is either empty or a hypersurface. An example is given to show that this is no longer true in the non-noetherian case.

We investigate an approach of Bass to study the Jacobian Conjecture via the degree of the inverse of a polynomial automorphism over an arbitrary ℚ-algebra.

The tame generators problem asked if every invertible polynomial map is tame, i.e. a finite composition of so-called elementary maps. Recently in [8] it was shown that the classical Nagata automorphism in dimension 3 is not tame. The proof is long and very technical. The aim of this paper is to present the main ideas of that proof.

Recently Umirbaev has proved the long-standing Anick conjecture, that is, there exist wild automorphisms of the free associative algebra $K\langle x,y,z\rangle$ over a field $K$ of characteristic 0. In particular, the well-known Anick automorphism is wild. In this article we obtain a stronger result (the Strong Anick Conjecture that implies the Anick Conjecture). Namely, we prove that there exist wild coordinates of $K\langle x,y,z\rangle$. In particular, the two nontrivial coordinates in the Anick automorphism are both wild. We establish a...

We study the group $\mathrm{Tame}\left({\mathrm{SL}}_2\right)$ of tame automorphisms of a smooth affine $3$-dimensional quadric, which we can view as the underlying variety of ${\mathrm{SL}}_2\left(ℂ\right)$. We construct a square complex on which the group admits a natural cocompact action, and we prove that the complex is $\mathrm{CAT}\left(0\right)$ and hyperbolic. We propose two applications of this construction: We show that any finite subgroup in $\mathrm{Tame}\left({\mathrm{SL}}_2\right)$ is linearizable, and that $\mathrm{Tame}\left({\mathrm{SL}}_2\right)$ satisfies the Tits alternative.

We discuss several additional properties a power linear Keller map may have. The Structural Conjecture of Drużkowski (1983) asserts that certain two such properties are equivalent, but we show that one of them is stronger than the other. We even show that the property of linear triangularizability is strictly in between. Furthermore, we give some positive results for small dimensions and small Jacobian ranks.

We present some estimates on the geometry of the exceptional value sets of non-zero constant Jacobian polynomial maps of ℂ² and their components.