# Counting number fields in fibers (with an Appendix by Jean Gillibert)

Counting number fields in fibers (with an Appendix by Jean Gillibert) 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 $${P_\tau \in X(\bar{\mathbb Q})}$$ P τ ∈ X ( Q ¯ ) such that $${t(P_\tau )=\tau }$$ t ( P τ ) = τ . Dvornicich and Zannier proved that, for large N, the field $${\mathbb Q}(P_1, \ldots , P_N)$$ Q ( P 1 , … , P N ) is of degree at least $$e^{cN/\log N}$$ e c N / log N 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Mathematische Zeitschrift Springer Journals

# Counting number fields in fibers (with an Appendix by Jean Gillibert)

, Volume 288 (2) – May 16, 2017
23 pages

/lp/springer_journal/counting-number-fields-in-fibers-with-an-appendix-by-jean-gillibert-ga9IYmvKKJ
Publisher
Springer Berlin Heidelberg
Copyright © 2017 by Springer-Verlag Berlin Heidelberg
Subject
Mathematics; Mathematics, general
ISSN
0025-5874
eISSN
1432-1823
D.O.I.
10.1007/s00209-017-1900-5
Publisher site
See Article on Publisher Site

### Abstract

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 $${P_\tau \in X(\bar{\mathbb Q})}$$ P τ ∈ X ( Q ¯ ) such that $${t(P_\tau )=\tau }$$ t ( P τ ) = τ . Dvornicich and Zannier proved that, for large N, the field $${\mathbb Q}(P_1, \ldots , P_N)$$ Q ( P 1 , … , P N ) is of degree at least $$e^{cN/\log N}$$ e c N / log N 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.

### Journal

Mathematische ZeitschriftSpringer Journals

Published: May 16, 2017

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