The main result of the paper is the following:
Let \(X\) be a smooth geometric irreducible algebraic curve of genus \(\mathbf{g}\) defined over a number field \(K\), \(d=[K:{\mathbb Q}]\) and \(t\in K(X)\) a non-constant rational function of degree \(n\ge 2\). For every \(\tau\in {\mathcal O}_K\), choose arbitrarily a point \(P_\tau \in t^{-1}(\tau)\). The there exist real numbers \(c=c(K,\mathbf g,n)>0\) and \(B_0=B_0(K,X,t)>1\) such that, for any \(B\ge B_0\), \[ [ K(P_\tau: \tau\in {\mathcal O}_K, H(\tau)\leq B)\,: K] \ge c^{cB^d/\log B}\, . \] As a corollary, one obtains that among the number fields \(K(P_\tau)\), where \(t\in{\mathcal O}\) and \(H(\tau)\leq B\), there are at least \(cB^d/\log B\) distinct fields.
This extends an analogous result of R. Dvornicich and U. Zannier [Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 21, No. 3, 421–443 (1994; Zbl 0819.12003)], where the field of definition of the curve \(X\) was just \({\mathbb Q}\).
The appendix by Jean Gillibert gives a scheme-theoretic explanation of the relation between geometric and arithmetic ramification of covers of \({\mathbb P}^1\), which is used to obtain important auxiliary results in order to prove the main theorem.