What is the importance of Jacobian Conjecture and any progress on it?

Question

What is the importance of Jacobian Conjecture?Are there any important central problem with the conjecture as precondition? and any progress on it?

Answer 1

This is a rather broad question and I'll try to give a bit of an overview of what I know. First of all, I'm not quite aware of central problems where the Jacobian Conjecture (JC) is a prerequisite, but there are several related, or even equivalent problems.

First of all, there's the cancellation problem: Let $k$ be an algebraically closed field of characteristic $0$. If $X$ is a $d$-dimensional variety over $k$ such that $X\times k^n\cong \mathbb{A}^{n+d}$, does it follow that $X\cong \mathbb{A}^d$? The techniques used to attack this problem are similar to techniques that are being used in relation to JC. In both cases, for instance, locally nilpotent (triangulable?) derivations are used.

Then there are several conjectures equivalent to JC, for instance the Dixmier and Mathieu conjectures. The Dixmier conjecture states that any endomorphism of the Weyl Algebra (algebra of polynomial differential operators) is invertible. See for instance here. The Mathieu conjecture states the following: Let $G$ be a compact, connected, real Lie group with Haar measure $\sigma$ and let $f$ be a complex-valued $G$-finite function over $G$ such that $\int_G fd\sigma=0$ for any $m\geq 1$, then for any $G$-finite function $g$ over $G$, $\int_Gf^mgd\sigma=0$, whenever $m\gg0$. (Link.)

When it comes to the progress on JC, several reductions have been made. A result of Bass-Connell-Wright (1982) shows that it is sufficient to prove JC for polynomial automorphisms of the form $F=(X_1+H_1,\ldots,X_n+H_n)$, where each $H_i$ is either zero, or a homogeneous cubic. Moreover, in this case invertibility of the Jacobian determanint is equivalent to $\mathcal{J}H$ being nilpotent. Van den Essen - De Bondt (2005) reduced this even further and showed that one may as well assume that $\mathcal{J}H$ is a symmetric matrix.

Currently, the research has developed in a somewhat different direction, as a result of some new conjectures by W. Zhao that are either equivalent to the Jacobian conjecture, or imply it. (Image conjecture, Vanishing conjecture, and a few others) Due to the similarity in structure between these conjectures and the previously mentioned Mathieu conjecture, this structure (introduced in this paper, as far as I can tell) is now being studied in the hopes that it reveals something new about JC.