Let $k$ be a field of characteristic zero (for example $\mathbb{R}$ or $\mathbb{C}$), and let $R=k[x_1,\ldots,x_n]$ be the $k$-algebra of polynomials in $n$ variables $x_1,\ldots,x_n$, $n \geq 1$.
(1) Does there exist a metric on $R$?
Let us consider the case $n=1$. In this case I think that we can define a metric as follows: For $A=\sum_{i=0}^{u}a_ix^i$ and $B=\sum_{i=0}^{v}b_ix^i$, we can think of $A=(a_0,\ldots,a_u,0,\ldots)$ and $B=(b_0,\ldots,b_v,0,\ldots)$, and define, in case $k \in \{\mathbb{R},\mathbb{C}\}$, for example: $d(A,B)=\sum |a_i-b_i|$.
Concerning $n \geq 2$, I think that again it is a metric space. For example, if we order the monomials as follows: $1,x,y,x^2,xy,^2,x^3,x^2y,xy^2,y^3,\ldots$, then each element of $k[x,y]$ can be thought of as a (finite) vector, and the distance between two elements is defined the same.
If I am not wrong, such a metric is not complete (and if we consider those metrices in $k[[x]]$/$k[[x,y]]$ etc., then they become complete).
So, if I am not wrong, I have answered my question, so I ask:
(2) Does there exist a complete metric on $R$?
Any hints and comments are welcome!
The polynomial rings $\Bbb{R}[x]$ and $\Bbb{C}[x]$ both have the same cardinality as $\Bbb{R}$ and hence can be made into complete metric spaces by picking a bijection with $\Bbb{R}$ and using it to transfer the usual metric on $\Bbb{R}$ to give an equivalent (and hence complete) metric on the polynomial ring. The resulting metric can't be expected to have some good properties you might want: e.g. there is no reason for the algebraic operations on the polynomial rings to be continuous.