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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?

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For the first question, see this question (and the link in the body of the question). For the second question, see the same question (and ESPECIALLY the discussion in the linked-in text,) as well as this question.

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