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I was just thinking that as rational functions form an ordered field you could describe analogous version of the absolute value function, but we don't quite have a 'metric' - for example |1/x| < e for all e > 0 in R but 1/x =/= 0.

I was wondering if anybody got anywhere with this 'metric' and if there are any links to papers exploring actual metrics on the rational functions?

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up vote 2 down vote accepted

The field $\mathbb R(x)$ of real rational functions is ordered by the condition that $ \frac{r_ox^n+...+r_n}{s_ox^m+...+s_m}>0$ if $r_0, s_o>0$.
This gives rise to a topology, which is metrizable:
The reason is that there is a denumerable set, consisting of the fractions $ \frac{1}{x^N}$, which is cofinal in the sense that for every positive rational real function$\frac{p(x)}{q(x)}>0$, there exists $N$ with $0< \frac{1}{x^N}<\frac{P(x)}{q(x)}$.
This implies that the ordered field $\mathbb R(x)$ is metrizable by a theorem of Dobbs that you can find here.

The whole paper is interesting and might serve as the reference you are looking for.

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