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Let $\mathbb{F}$ be a field. And let $\mathbb{F}(\bar x)$ be the field of rational functions over the field $\mathbb{F}$ in the indeterminate $x_1,\dots,x_n$. Namely, all the functions $f:\mathbb{F}^n\rightarrow \mathbb{F}$ such that $f$ can be written as $f(\bar x)=q(\bar x)/p(\bar x)$ for two formal polynomials $q,p\in\mathbb{F}[\bar x]$.

Question: is the representation of $f(\bar x)$ as $q(\bar x)/p(\bar x)$ unique? Namely, can we have two different pairs of $q,p$ that represent the same function $f$. I am interested both in the case that the field $\mathbb{F}$ is infinite, or finite.

Jack
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  • By https://math.stackexchange.com/questions/4202602/when-are-rational-functions-considered-equal-eg-fracz-1z2-1-vs-frac it seems that it depends on the definition for rational functions. But my question is indeed what is the canonical definition of rational functions? Is there one? – Jack Dec 27 '21 at 16:16
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    There is always nonuniqueness in that you can multiply both $f$ and $g$ by the same nonzero polynomial. But by unique factorization in polynomial rings over a field, that's the only nonuniqueness. – Robert Israel Dec 27 '21 at 16:21
  • I see, thanks. So the canonical definition is that there is NO uniqueness. Note that in areas like theoretical computer science, this may make a difference, because polynomials are considered as syntactic terms. – Jack Dec 27 '21 at 16:22
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    @jack The canonical definition is in terms of equivalence classes. Same as in constructing $\Bbb Q$ from $\Bbb Z$. – Hagen von Eitzen Dec 27 '21 at 16:44
  • @HagenvonEitzen, thanks. So there IS uniqueness in this sense, but one can choose any representative for each equivalence class. Again, note that in theoretical computer science these seemingly non-essential matters, may be important. That's why I'm asking. – Jack Dec 27 '21 at 16:53

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