I'm in fact dealing with Exercise 4.13(d) of Gathmann's 2021 notes of Algebraic Geometry.
The Exercises asks that if $f:\mathbb{A}^2\to \mathbb{A}^2$ is an isomorphism of affine varieties, then prove or disprove that $f$ is affine linear, i.e. it is of the form $f(x)=Ax+b$ for some $A\in \text{Mat}(2\times 2,K)$ and $b\in K^2$. ($K$ is assumed to be algebraically closed.)
A friend told me the following result which is proved in this paper:
If $F:\mathbb{C}^n\to \mathbb{C}^n$ is a polynomial map which is one-to-one, then
(a) $F(\mathbb{C}^n)=\mathbb{C}^n$, and
(b) $F^{-1}\colon \mathbb{C}^n\to \mathbb{C}^n$ is also a polynomial map.
Therefore if I could find a non-linear injective polynomial map from $\mathbb{C}^2\to \mathbb{C}^2$ (and better if I could write out its polynomial inverse), then I would be done disproving the statement.
However, naive try on $n=1$ cannot be helpful: an injective polynomial map $\mathbb{C}\to\mathbb{C}$ must be of degree $1$ since $\mathbb{C}$ is algebraically closed and the number of roots of a single-variable polynomial equals its degree.
And $n\geq 2$ seems too not naive for me...
I think there should be some maybe-well-known examples of non-linear injective polynomial maps for $n\geq 2$, since if it can be proved (or even conjectured without any counterexample) that an injective polynomial map must be linear, then the result above would be trivial linear algebra.
Unfortunately, the paper above does not have any example for this, and I just cannot come up with a such example.
Thanks in advance for any help.
