# Integral values of rational map

This question is related to this post on Math.MO.

A theorem of B.Segre tells us that if there is one rational point on a non-singular cubic surface $X$ over $\mathbb{Q}$, then the surface is unirational, i.e., there is a dominant rational map from $\mathbb{P}^2$ to $X$ over $\mathbb{Q}$.

Question: 1)Suppose $f(t)$ is a rational function over $\mathbb{Q}$. Is there an algorithm for finding out whether there is such a $t_0\in\mathbb{Q}$ that $f(t_0)\in\mathbb{Z}$?

2)Suppose $u(x,y),v(x,y)$ are two rational functions over $\mathbb{Q}$. Is there an algorithm for finding out whether there is such a pair $(x_0,y_0)\in\mathbb{Q}^2$ that $u(x_0,y_0)\in\mathbb{Z}$ and $v(x_0,y_0)\in\mathbb{Z}$ simultaneously?