An rr function (i.e. *rational rational function*) is a quotient


$$ \frac fg\,:\, \Bbb Q\ \to\ \Bbb Q\cup\{\infty\} $$


such that $\ f,g\,\in\,\Bbb Z[X],\ $ where $\ g\ne 0.$


**QUESTION** Do there exist rr functions $\ \phi\ \psi\ $
such that set


$$ \{(\phi(x)\ \ \psi(x))\,:\, x\in\Bbb Q\}\ \subseteq
 \ (\Bbb Q\cup\{\infty\})^2 $$


is dense in a non-empty open subset of
$\ \Bbb Q^2\,?$


I'd guess -- **NO**.



Rational Peano curves
Use the resultant to eliminate the variable $X$. Since the resultant is computed over the rationals, the resultant is a rational coefficient polynomial in the two variables of the plane, satisfied on the image of the parameterized curve. See my (undergraduate!) lecture notes [Concrete Algebra](https://github.com/Ben-McKay/concrete-algebra/blob/master/algebra.pdf) on github for complete (and elementary) details. 


