LAUSR

Let X be a projective curve over $${\mathbb Q}$$Q and $${t\in {\mathbb Q}(X)}$$t∈Q(X) a non-constant rational function of degree $${n\ge 2}$$n≥2. For every $${\tau \in {\mathbb Z}}$$τ∈Z pick $${P_\tau \in X(\bar{\mathbb Q})}$$Pτ∈X(Q¯) such that $${t(P_\tau )=\tau }$$t(Pτ)=τ. Dvornicich and Zannier proved that, for large N, the field $${\mathbb Q}(P_1, \ldots , P_N)$$Q(P1,…,PN) is of degree at least $$e^{cN/\log N}$$ecN/logN over $${\mathbb Q}$$Q, where $${c>0}$$c>0 depends only on X and t. In this paper we extend this result, replacing $${\mathbb Q}$$Q by an arbitrary number field.