We consider nonsingular polynomial maps F = (P,Q): ℝ² → ℝ² under the following regularity condition at infinity $\left(J_{\infty }\right)$: There does not exist a sequence $(p_k,q_k)\subset ℂ²$ of complex singular points of F such that the imaginary parts $(\Im \left(p_k\right),\Im \left(q_k\right))$ tend to (0,0), the real parts $(\Re \left(p_k\right),\Re \left(q_k\right))$ tend to ∞ and $F(\Re \left(p_k\right),\Re \left(q_k\right)))\to a\in ℝ²$. It is shown that F is a global diffeomorphism of ℝ² if it satisfies Condition $\left(J_{\infty }\right)$ and if, in addition, the restriction of F to every real level set $P^{-1}\left(c\right)$ is proper for values of |c| large enough.

We show that the GVC (generalized vanishing conjecture) holds for the differential operator $\Lambda =({\partial }_x-\Phi \left({\partial }_y\right)){\partial }_y$ and all polynomials $P(x,y)$, where $\Phi \left(t\right)$ is any polynomial over the base field. The GVC arose from the study of the Jacobian conjecture.

We associate to a given polynomial map from ${ℂ}^2$ to itself with nonvanishing Jacobian a variety whose homology or intersection homology describes the geometry of singularities at infinity of this map.

Given a real n×n matrix A, we make some conjectures and prove partial results about the range of the function that maps the n-tuple x into the entrywise kth power of the n-tuple Ax. This is of interest in the study of the Jacobian Conjecture.

In certain cases the invertibility of a polynomial map F = (P,Q): ℂ²→ ℂ² can be characterized by the irreducibility and the rationality of the curves aP+bQ = 0, (a:b) ∈ ℙ¹.

A non-zero constant Jacobian polynomial map F=(P,Q):ℂ² → ℂ² has a polynomial inverse if the component P is a simple polynomial, i.e. its regular extension to a morphism p:X → ℙ¹ in a compactification X of ℂ² has the following property: the restriction of p to each irreducible component C of the compactification divisor D = X-ℂ² is of degree 0 or 1.

It is shown that the invertible polynomial maps over a finite field Fq , if looked at as bijections Fn,q −→ Fn,q , give all possible bijections in the case q = 2, or q = p^r where p > 2. In the case q = 2^r where r > 1 it is shown that the tame subgroup of the invertible polynomial maps gives only the even bijections, i.e. only half the bijections. As a consequence it is shown that a set S ⊂ Fn,q can be a zero set of a coordinate if and only if #S = q^(n−1).

Soit $\phi :=(f,g):{ℂ}^2\to {ℂ}^2$ où $f$ et $g$ sont des applications polynomiales. Nous établissons le lien qui existe entre le polygone de Newton de la courbe réunion du discriminant et du lieu de non-propreté de $\phi$ et la topologie des entrelacs à l’infini des courbes affines $f^{-1}\left(0\right)$ et $g^{-1}\left(0\right)$. Nous en déduisons alors des conséquences liées à la conjecture du jacobien.

We describe some recent developments concerning the Jacobian Conjecture (JC). First we describe Drużkowski’s result in [6] which asserts that it suffices to study the JC for Drużkowski mappings of the form $x+{\left(Ax\right)}^{*3}$ with A² = 0. Then we describe the authors’ result of [2] which asserts that it suffices to study the JC for so-called gradient mappings, i.e. mappings of the form x - ∇f, with $f\in k^{\left[n\right]}$ homogeneous of degree 4. Using this result we explain Zhao’s reformulation of the JC which asserts the following:...

Implementations of known reductions of the Strong Real Jacobian Conjecture (SRJC), to the case of an identity map plus cubic homogeneous or cubic linear terms, and to the case of gradient maps, are shown to preserve significant algebraic and geometric properties of the maps involved. That permits the separate formulation and reduction, though not so far the solution, of the SRJC for classes of nonsingular polynomial endomorphisms of real n-space that exclude the Pinchuk counterexamples to the SRJC,...

Existence of loops for non-injective regular analytic transformations of the real plane is shown. As an application, a criterion for injectivity of a regular analytic transformation of ${ℝ}^2$ in terms of the Jacobian and the first and second order partial derivatives is obtained. This criterion is new even in the special case of polynomial transformations.

We show that the plane Jacobian conjecture is equivalent to finite generatedness of certain modules.

In the recent work [BE1], [Me], [Burgers] and [HNP], the well-known Jacobian conjecture ([BCW], [E]) has been reduced to a problem on HN (Hessian nilpotent) polynomials (the polynomials whose Hessian matrix is nilpotent) and their (deformed) inversion pairs. In this paper, we prove several results on HN polynomials, their (deformed) inversion pairs as well as on the associated symmetric polynomial or formal maps. We also propose some open problems for further study.

Nel presente lavoro si studiano le applicazioni polinomiali proprie $$\phi :{\mathbb{R}}^n\to {\mathbb{R}}^q.$$ In particolare si prova: 1) se $\phi :{\mathbb{R}}^n\to \mathbb{R}$ è un'applicazione polinomiale tale che ${\phi }^{-1}\left(y\right)$ è compatto per ogni $y\in \mathbb{R}$, allora $\phi$ è propria; 2) se $\phi :{\mathbb{R}}^n\to {\mathbb{R}}^q$ è polinomiale a fibra compatta e $\phi \left({\mathbb{R}}^n\right)$ è chiuso in ${\mathbb{R}}^q$ allora $\phi$ è propria; 3) l'insieme delle applicazioni polinomiali proprie di ${\mathbb{R}}^n$ in ${\mathbb{R}}^q$ è denso, nella topologia $C^{\mathrm{\infty }}$, nello spazio delle applicazioni $C^{\mathrm{\infty }}$ di ${\mathbb{R}}^n$ in ${\mathbb{R}}^q$.