Suppose $f(x_1,x_2,\dots)=\frac{P}{Q}$, where $P,Q$ are polynomials in several variables with integer coefficients that have the same degree. Let's denote by $S(f)$ the set of integers $n$ for which $f(x_1,x_2,\dots)=n$ is solvable in integers.

Which sets $S\subset \mathbb Z$ can be written as $S(f)$ for some $f$ as above?

For example we have, $S(\frac{x_1^2+x_2^2}{x_1x_2+1})=\lbrace -5,0,1,4,\dots,k^2,\dots\rbrace$.

This question is just a musing from playing around with variations to Hilbert's tenth problem. A more direct question would be: Is every Diophantine set representable as some $S(f)$?