This article is about polynomial maps with a certain symmetry and/or antisymmetry in their Jacobians, and whether the Jacobian Conjecture is satisfied for such maps, or whether it is sufficient to prove the Jacobian Conjecture for such maps. For instance, we show that it suffices to prove the Jacobian conjecture for polynomial maps x + H over ℂ such that satisfies all symmetries of the square, where H is homogeneous of arbitrary degree d ≥ 3.

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.

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

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.

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.

Let (F₁,..., Fₙ): ℂⁿ → ℂⁿ be a locally invertible polynomial map. We consider the canonical pull-back vector fields under this map, denoted by ∂/∂F₁,...,∂/∂Fₙ. Our main result is the following: if n-1 of the vector fields $\partial /\partial F_j$ have complete holomorphic flows along the typical fibers of the submersion $(F₁,...,F_{j-1},F_{j+1},...,Fₙ)$, then the inverse map exists. Several equivalent versions of this main hypothesis are given.

Let d be any integer greater than or equal to 3. We show that the intersection of the set mdeg(Aut(ℂ³))∖ mdeg(Tame(ℂ³)) with {(d₁,d₂,d₃) ∈ (ℕ ₊)³: d = d₁ ≤ d₂≤ d₃} has infinitely many elements, where mdeg h = (deg h₁,...,deg hₙ) denotes the multidegree of a polynomial mapping h = (h₁,...,hₙ): ℂⁿ → ℂⁿ. In other words, we show that there are infinitely many wild multidegrees of the form (d,d₂,d₃), with fixed d ≥ 3 and d ≤ d₂ ≤ d₃, where a sequence (d₁,...,dₙ)∈ ℕ ⁿ is a wild multidegree if there is...