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

Kaliada, Dzianis; Götze, Friedrich; Kukso, Olga

doi:10.1007/s10986-014-9234-z

Cited by 7 publications

(4 citation statements)

References 16 publications

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“…Remark. The corresponding results for n = 2 and n = 3 are obtained in [12] and [20]. Theorem 2 shows that lim x→+0 φ n (x) = +∞.…”

Section: 3mentioning

confidence: 76%

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On the Distribution of Polynomial Discriminants: Totally Real Case

Koleda

2019

Lith Math J

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In the paper we study the distribution of the discriminant D(P ) of polynomials P from the class P n (Q) of all integer polynomials of degree n and height at most Q. We evaluate the asymptotic number of polynomials P ∈ P n (Q) having all the roots real and satisfying the inequality |D(P )| ≤ X as Q → ∞ and X/Q 2n−2 → 0. Brief outlineSection 1 contains the background of the subject and the main results of the paper. Section 2 includes auxiliary proposions necessary to prove the main results; the reader can skip this section in the first reading. In Section 3 we express the distribution function of the discriminant via the volumes of specific regions. In Section 4 we study the asymptotic behaviour of the corresponding volume for small values of discriminant of polynomials having all the roots real.Everywhere in the paper the degree n ≥ 4 of polynomials is fixed; the parameter Q bounding heights of polynomials from above grows to infinity.

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“…Remark. The corresponding results for n = 2 and n = 3 are obtained in [12] and [20]. Theorem 2 shows that lim x→+0 φ n (x) = +∞.…”

Section: 3mentioning

confidence: 76%

“…The upper bounds for degree n = 3 one can find in [21], [20], [3]; on the asymptotics for n = 2 see [12].…”

Section: 3mentioning

confidence: 99%

On the Distribution of Polynomial Discriminants: Totally Real Case

Koleda

2019

Lith Math J

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“…Using the last estimate, (17) and the fact that the number of different vectors b 0 does not exceed (2Q + 1) n , we get…”

Section: Large Derivativementioning

confidence: 98%

Quantitative Estimate for the Measure of a Set of Real Numbers

Budarina¹

2021

Glasgow Math. J.

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“…However, for n = 3 and ν ∈ [0, 3/5) Kaliada, Götze, and Kukso [11] obtained the following asymptotic relation:…”

Section: Introductionmentioning

confidence: 99%

Discriminant and Root Separation of Integral Polynomials

Götze

Zaporozhets

2016

J Math Sci

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Consider a random polynomial G Q (x) = ξ Q,n x n + ξ Q,n−1 x n−1 + · · · + ξ Q,0 with independent coefficients uniformly distributed on 2Q + 1 integer points {−Q, . . . , Q}. Denote by D(G Q ) the discriminant of G Q . We show that there exists a constant Cn, depending on n only such that for all Q ≥ 2 the distribution of D(G Q ) can be approximated as followswhere ϕn denotes the distribution function of the discriminant of a random polynomial of degree n with independent coefficients which are uniformly distributed on [−1, 1]. Let ∆(G Q ) denote the minimal distance between the complex roots of G Q . As an application we show that for any ε > 0 there exists a constant δn > 0 such that ∆(G Q ) is stochastically bounded from below/above for all sufficiently large Q in the following sense P δn < ∆(G Q ) < 1 δn > 1 − ε.

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

Cited by 7 publications

References 16 publications

On the Distribution of Polynomial Discriminants: Totally Real Case

On the Distribution of Polynomial Discriminants: Totally Real Case

Quantitative Estimate for the Measure of a Set of Real Numbers

Discriminant and Root Separation of Integral Polynomials

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