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 ${\mathbb{R}}^q$.

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.