Given a field $k$, consider the polynomial ring $k[x_1,x_2,\dots,x_n]$. Is it possible to find all the automorphisms of this ring that fix the field $k$?
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Let $X = (x_1, x_2, \ldots , x_n)$. Automorphisms are polynomial substitutions $f(X) \to f(P(X))$ where $P$ has a polynomial inverse.
Structure theory of the automorphism group is complicated for $n \geq 3$. A recent breakthrough was the proof that not all of the group is generated by substitutions $x_i \to ax_i +b$ where $a$ is a constant and $b$ is a polynomial in the other variables.
For $n=2$ these substitutions (called tame automorphisms) generate the full automorphism group and the structure of the group is explained in Cohn's book on free rings. The structure of the tame subgroup for $n=3$ is determined in
This is all very closely connected to the Jacobian conjecture.