Diversity in Parametric Families of Number Fields

Yuri Bilu () and Florian Luca ()
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Yuri Bilu: Université de Bordeaux & CNRS, Institut de Mathématiques de Bordeaux
Florian Luca: Wits University, School of Mathematics

A chapter in Number Theory – Diophantine Problems, Uniform Distribution and Applications, 2017, pp 169-191 from Springer

Abstract: Abstract Let X be a projective curve defined over ℚ $$\mathbb{Q}$$ and t ∈ ℚ ( X ) $$t \in \mathbb{Q}(X)$$ a non-constant rational function of degree ν ≥ 2. For every n ∈ ℤ $$n \in \mathbb{Z}$$ pick P n ∈ X ( ℚ ̄ ) $$P_{n} \in X(\bar{\mathbb{Q}})$$ such that t(P n ) = n. A result of Dvornicich and Zannier implies that, for large N, among the number fields ℚ ( P 1 ) , … , ℚ ( P N ) $$\mathbb{Q}(P_{1}),\ldots, \mathbb{Q}(P_{N})$$ there are at least cN∕ logN distinct; here, c > 0 depends only on the degree ν and the genus g = g(X). We prove that there are at least N∕(logN)1−η distinct fields, where η > 0 depends only on ν and g.

Keywords: Zannier; Square-free Numbers; Finite Critical Value; Square-free Positive Integer; Hilbert's Irreducibility Theorem (search for similar items in EconPapers)
Date: 2017
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DOI: 10.1007/978-3-319-55357-3_7

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