Diversity in rationally parameterized number fields

Let X be a curve defined over Q and let t ∈ Q(X) be a non-constant rational function on X of degree v ≥ 2. For every rational number a/b pick a point P_a/b ∈ X(Ǭ) such that t(P_a/b) = a/b. In this paper, we obtain lower bounds on the number of distinct fields among Q(P_a/b) with 1 ≤ a, b ≤ N under some assumptions on t. We show that if t has a pole of order at least 2 or if there is a rational number α such that t − α has a zero of order at least 2, then the set {Q(P_a/b) | 1 ≤ a, b ≤ N} contains (Figure presented)elements. We also obtain partial results when t does not have a pole of order at least two.