Counting Number Fields in Fibers
Number Theory
2017-04-03 v2
Abstract
Let X be a projective curve over Q and t a non-constant Q-rational function on X of degree n>1. For every integer a pick a points P(a) on X such that t(P(a))=a. Dvornicich and Zannier (1994) proved that for large N the field Q(P(1), ..., P(N)) is of degree at least exp(cN/log N) over Q, where c>0 depends only on X and t. In this note we extend this result, replacing Q by an arbitrary number field.
Cite
@article{arxiv.1606.02341,
title = {Counting Number Fields in Fibers},
author = {Yuri Bilu and Jean Gillibert},
journal= {arXiv preprint arXiv:1606.02341},
year = {2017}
}
Comments
Minor inaccuracies corrected following the referee's suggestions. To appear in Math. Z