Abstract
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 Pa/b ∈ X(Ǭ) such that t(Pa/b) = a/b. In this paper, we obtain lower bounds on the number of distinct fields among Q(Pa/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(Pa/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.
Original language | English |
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Pages (from-to) | 2353-2364 |
Number of pages | 12 |
Journal | International Journal of Number Theory |
Volume | 19 |
Issue number | 10 |
DOIs | |
Publication status | Published - 1 Nov 2023 |
Keywords
- Algebraic curves
- Diophantine equations
- Field arithmetic
- Galois theory
ASJC Scopus subject areas
- Algebra and Number Theory