Lithuanian Mathematical Journal

, Volume 54, Issue 2, pp 150–165 | Cite as

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

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 Q14/5XQ4 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.

Keywords

polynomial discriminant cubic polynomial distribution of discriminants 

MSC

11J25 11J83 

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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Institute of MathematicsNational Academy of Sciences of BelarusMinskBelarus
  2. 2.University of BielefeldBielefeldGermany

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