I was just thinking that as rational functions form an ordered field you could describe...

Jamiya Weber

Jamiya Weber

Answered

2022-06-25

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?

Answer & Explanation

Samantha Reid

Samantha Reid

Expert

2022-06-26Added 22 answers

The field R ( x ) of real rational functions is ordered by the condition that r o x n + . . . + r n s o x 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 1 x N , which is cofinal in the sense that for every positive rational real function p ( x ) q ( x ) > 0, there exists N with 0 < 1 x N < P ( x ) q ( x ) .
This implies that the ordered field R ( x ) is metrizable by a theorem of Dobbs.
The whole paper is interesting and might serve as the reference you are looking for.

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