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The asymptotic number of integral cubic polynomials with bounded heights and discriminants

Lithuanian Mathematical Journal volume 54, pages 150–165 (2014)Cite this article

Abstract

Let P denote a cubic integral polynomial, and let D(P) and H(P) denote the discriminant and height of P, respectively. Let N(Q,X) be the number of cubic integral polynomials P such that H(P) ≤ Q and |D(P)| ≤ X. We obtain an asymptotic formula of N(Q,X) for Q 14/5 ≪ X ≪ Q 4 and Q → +∞. Using this result, for 0 ≤ η ≤ 9/10, we find the asymptotic value of

$$ \sum\limits_{{\begin{array}{*{20}{c}} {H(P)\leq Q} \\ {1\leq \left| {D(P)} \right|\ll {Q^{{4-\eta }}}} \\ \end{array}}} {{{{\left| {D(P)} \right|}}^{{-{1 \left/ {2} \right.}}}}}, $$

where the sum is taken over irreducible integral polynomials and Q → +∞. This improves upon a result of Davenport, who dealt with the case η = 0.

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Authors and Affiliations

  1. Institute of Mathematics, National Academy of Sciences of Belarus, 220072, Minsk, Belarus

    Dzianis Kaliada & Olga Kukso

  2. University of Bielefeld, 33501, Bielefeld, Germany

    Friedrich Götze

Authors
  1. Dzianis Kaliada
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  2. Friedrich Götze
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  3. Olga Kukso
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Correspondence to Dzianis Kaliada.

Additional information

Dzianis Kaliada and Olga Kukso would like to thank the University of Bielefeld, where a substantial part of this work was done, for providing a stimulating research environment during his visit supported by SFB 701.

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Kaliada, D., Götze, F. & Kukso, O. The asymptotic number of integral cubic polynomials with bounded heights and discriminants. Lith Math J 54, 150–165 (2014). https://doi.org/10.1007/s10986-014-9234-z

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  • DOI: https://doi.org/10.1007/s10986-014-9234-z

MSC

  • 11J25
  • 11J83

Keywords

  • polynomial discriminant
  • cubic polynomial
  • distribution of discriminants