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

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## Abstract

Let *X* be a projective curve over \({\mathbb Q}\) and \({t\in {\mathbb Q}(X)}\) a non-constant rational function of degree \({n\ge 2}\). For every \({\tau \in {\mathbb Z}}\) pick \({P_\tau \in X(\bar{\mathbb Q})}\) such that \({t(P_\tau )=\tau }\). Dvornicich and Zannier proved that, for large *N*, the field \({\mathbb Q}(P_1, \ldots , P_N)\) is of degree at least \(e^{cN/\log N}\) over \({\mathbb Q}\), where \({c>0}\) depends only on *X* and *t*. In this paper we extend this result, replacing \({\mathbb Q}\) by an arbitrary number field.

## Notes

### Acknowledgements

A substantial part of this article was written during my stay the Max-Planck-Institut für Mathematik in Bonn. I thank this institute for the financial support and stimulating working conditions.

This article belongs to a joint project with Jean Gillibert. I thank him for allowing me to publish this part of this project as a separate article, for adding a beautiful appendix, and for many stimulating discussions.

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