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

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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.

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  • $\begingroup$ And then I of course forgot to mention the book that was a more or less complete overview of the state of affairs in 2000: "Polynomial Automorphisms and the Jacobian Conjecture" by A. van den Essen, published by Birkhäuser Verlag. $\endgroup$
    – HSN
    Sep 14, 2014 at 14:59
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    $\begingroup$ In Nice there were a flow of talks by invited speakers who claimed to have solved the Jacobian conjecture. The procedure was that Joël Briançon, a brilliant specialist who had thought a lot about it, would sit in the audience and point at some time in the talk to a dubious assertion. The speaker would say that he would give the details later but of course he was never able to give a correct argument. It seems that there are many flawed proofs of the Jacobian conjecture in which the mistake is diabolically hard to spot. $\endgroup$ Sep 16, 2014 at 8:04
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    $\begingroup$ There are a lot of incorrect proofs for JC, indeed, and every now and then a new one emerges. More about this can be found here: sbseminar.wordpress.com/2009/05/27/…. $\endgroup$
    – HSN
    Sep 17, 2014 at 11:44

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