Diversity in Parametric Families of Number Fields

Chapter

Abstract

Let X be a projective curve defined over \(\mathbb{Q}\) and \(t \in \mathbb{Q}(X)\) a non-constant rational function of degree ν ≥ 2. For every \(n \in \mathbb{Z}\) pick \(P_{n} \in X(\bar{\mathbb{Q}})\) such that t(P n ) = n. A result of Dvornicich and Zannier implies that, for large N, among the number fields \(\mathbb{Q}(P_{1}),\ldots, \mathbb{Q}(P_{N})\) 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.

Notes

Acknowledgements

During the work on this article Yuri Bilu was partially supported by the University of Xiamen, and by the binational research project MuDeRa, funded jointly by the French ANR and the Austrian FWF. Florian Luca was supported by an A-rated researcher award of the NRF of South Africa.

We thank Jean Gillibert and Felipe Voloch for useful discussions. We also thank the referees who carefully read the manuscript and detected several inaccuracies.

References

  1. 1.
    Yu. Bilu, Counting Number Fields in Fibers (With An Appendix by Jean Gillibert). Math. Z. (2017, to appear). arXiv:1606.02341[math.NT]Google Scholar
  2. 2.
    Yu. Bilu, F. Luca, Number Fields in Fibers: The Geometrically Abelian Case with Rational Critical Values. Periodica Math. Hung. (to appear). arXiv:1606.09164[math.NT]Google Scholar
  3. 3.
    P. Corvaja, U. Zannier, On the number of integral points on algebraic curves. J. Reine Angew. Math. 565, 27–42 (2003)MathSciNetMATHGoogle Scholar
  4. 4.
    H. Davenport, D. Lewis, A. Schinzel, Polynomials of certain special types. Acta Arith. 9, 107–116 (1964)MathSciNetMATHGoogle Scholar
  5. 5.
    J.-M. De Koninck, F. Luca, Analytic Number Theory: Exploring the Anatomy of Integers. Graduate Studies in Mathematics, vol. 134 (AMS, Providence, RI, 2012)Google Scholar
  6. 6.
    R. Dvornicich, U. Zannier, Fields containing values of algebraic functions. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 21, 421–443 (1994)Google Scholar
  7. 7.
    R. Dvornicich, U. Zannier, Fields containing values of algebraic functions II (On a conjecture of Schinzel). Acta Arith. 72, 201–210 (1995)MathSciNetMATHGoogle Scholar
  8. 8.
    J.-P. Serre, Lectures on the Mordell-Weil Theorem, 3rd edn. (Vieweg & Sohn, Braunschweig, 1997)CrossRefMATHGoogle Scholar
  9. 9.
    U Zannier, On the number of times a root of f(n, x) = 0 generates a field containing a given number field. J. Number Theory 72, 1–12 (1998)Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Institut de Mathématiques de BordeauxUniversité de Bordeaux & CNRSTalenceFrance
  2. 2.School of MathematicsWits UniversityJohannesburgSouth Africa

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