Let $k[x,y]$ be the polynomial ring in two variables over a field $k$ of characteristic zero. Every $k$-algebra automorphism of $k[x,y]$ is tame (e.g. the paper of McKay and Wang). It was pointed out in an answer to [this question](https://mathoverflow.net/questions/228657/is-a-wild-automorphism-of-kx-1-ldots-x-n-n-geq-3-necessarily-of-infini) that $k[x,y,z]$ has wild automorphisms of finite order.


My question is, is every automorphism of $k[x,y]$ of finite order a composition of elementary automorphisms of type (ii) as in the abstract of McKay and Wang? My rough thinking is that elementary automorphisms of type (i) increase total degree and therefore cannot be finite order, but the example of the wild automorphism of finite order for $k[x,y,z]$ has me questioning this reasoning.


EDIT: I would also be happy with a similar statement about finite order automorphisms of $k[x]$, if $k[x,y]$ is too ambitious.


*McKay, James H.; Wang, Stuart Sui-Sheng*, [**An elementay proof of the automorphism theorem for the polynomial ring in two variables**](http://dx.doi.org/10.1016/0022-4049(88)90137-5), J. Pure Appl. Algebra 52, No. 1-2, 91-102 (1988). [ZBL0656.13002](https://zbmath.org/?q=an:0656.13002).



Is a finite order automorphism of k[x,y] necessarily linear?
The answer seems to be no. The automorphism group of $k[x,y]$ can be described as the amalgamated product of the affine transformations with the group of maps of type (i) as in the abstract of McKay and Wang. The torsion elements in an amalgamated product are those elements conjugate to a torsion element of one of the factors, so the torsion elements of $\operatorname{Aut}(k[x,y])$ are those conjugate to affine transformations. This includes many automorphisms which are not affine.


