Let $k$ be a field of characteristic zero (I do not mind to assume that $k \in \{\mathbb{R},\mathbb{C}\}$), and let $R=k[x,y]$ be the $k$-algebra of polynomials in two variables $x,y$. Let $f$ be a $k$-algebra endomorphism of $R$.
(I) Is there a fixed-point theorem that can be applied to $f$? (II) If so, then is it possible to obtain a fixed point $\in R-k$? (the elements of $k$ are always fixed points for a $k$-algebra endomorphism $f$).
Remarks:
(1) I have tried to consider $k[x,y]$ as a complete metric space, see this question, but I guess that $f$ is not a contraction in that metric. So, is there another metric that makes $f$ a contraction? (probably yes? But it may be difficult to guarantee that $f$ has a fixed point $\in R-k$).
(2) The answer probably depends on the given $f$, since, for example, it seems that $f:(x,y) \mapsto (x^2,y^2)$ does not have fixed points other than the elements of $k$.
(3) Here is a list of fixed point theorems.
(4) The following questions seem relevant: a and b (what if we consider our given endomorphism as a multiplicative group morphism?).
(5) A similar question of mine is this (the fourth remark there says that the existence of such a fixed point implies the two-dimensional Jacobian Conjecture).
Thank you very much!
I very much doubt you can approach it in this fashion.
If there is a fixed point, then the fixed set would have to be a subalgebra, so in particular, a vector subspace, and this endomorphism of course respects this structure so you probably want a metric that also respects this.
In particular, you want proper vector subspaces to be closed and nowhere dense. But this yields a contradiction when you demand the metric to be complete, because we can write $k[x,y]$ as a countable union of proper (hence closed nowhere dense) subspaces $$ k[x,y]=\bigcup_{d=0}^\infty \operatorname{span}_k\{x^iy^j\mid i+j\leq d\} $$ but Baire category theorem says a complete metric space is not meagre.
A few words about the formal power series ring
If you decide to work with the formal power series ring $k[[x,y]]$ instead, then obviously your endomorphism $f$ must send the nilpotent $x,y$ to some nilpotent elements. This means in particular that we can reduce to looking at fixed elements of each $(x,y)/(x,y)^m$ and patching each graded piece together, like in Hansel's lemma.