Algebraic functions with infinitely many values in a number field

Abstract

We describe algebraic curves $X:F(x,y)=0$ defined over $\overline{\mathbb{Q}}$ that satisfy the following property: there exist a number field $k$ and an infinite set $S\subset k$ such that, for every $y\in S$, the roots of the polynomial $F(x,y)$ belong to $k$.

An extended and polished version