Bilu–Luca conjecture on distinct number fields in fibers

Let $X$ be a smooth geometrically irreducible projective algebraic curve over ${\mathbb Q}$, and let $t\in {\mathbb Q}(X)$ be a non-constant ${\mathbb Q}$-rational function such that

$${\mathbb Q}(X)\ne {\mathbb Q}(t).$$

For every $n\in {\mathbb Z}$, choose $P_n\in X(\overline{\mathbb Q})$ with $t(P_n)=n$.

Bilu–Luca conjecture. There exists a real number $c>0$ such that, for every sufficiently large integer $N$, at least $cN$ of the number fields ${\mathbb Q}(P_1),\ldots,{\mathbb Q}(P_N)$ are distinct.

This strengthens the known lower bound of order $N/\log N$ for the number of distinct fields, obtained from a lower bound on the degree of ${\mathbb Q}(P_1,\ldots,P_N)$. The conjecture is proved in the paper when ${\overline{\mathbb Q}}(X)/{\overline{\mathbb Q}}(t)$ is abelian and all critical values of $t$ are rational; in general, it is presented as following from a conjecture of Schinzel.