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Here $A_1 := K\{x\cdot-, \frac{d}{dx}\} \subset \operatorname{End}_K(K[x])$ for some characteristic-zero field $K$.

I found this claim in Coutinho's "A Primer of Algebraic D-Modules." If this is true for arbitrary $n$ it implies the Jacobian conjecture, but of course the Jacobian conjecture is trivial when $n = 1$.

If this is really open, why is it a hard problem?

Daniel McLaury
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    I'm not sure if you can see my deleted answer. I never intended to submit that as an answer to anything, much less as an answer to your question. (I was having trouble getting the align environment to compile like I wanted and wanted to try it out on a different compiler to see how it looked). I didn't even realize I had submitted it till I randomly checked M.SE and saw several notifications. I'm quite embarrassed and incredibly sorry for wasting your time with it. Apologies to you, and to all the wonderful commentors who notified me of the issue! – Jason DeVito - on hiatus Jan 30 '14 at 01:42

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I am sorry for non-useful answer.

I think, that the best way to understand why the question is hard is to try solving it. Even finding automorphisms of $A_1$ is a complicated problem. It was done by Dixmier (sur les alberes de weyl) and the basic thing is that you can describe automorphisms of polynomial ring in two variables.

The relation with Jacobian conjecture is more intresting: to prove this conjecture by using Jacobian conjecture you need to prove 2-Jacobian conjecture.

Here you can find the description of automorphisms of $A_1$ and maybe it will give you some understanding why this question is nontrivial.

user68061
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