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
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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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
- polynomial discriminant
- cubic polynomial
- distribution of discriminants