On the divisibility of the discriminant of an integral polynomial by prime powers

Bernik, V. I.; Götze, Friedrich; Kukso, Olga

doi:10.1007/s10986-008-9025-5

Cited by 10 publications

(5 citation statements)

References 7 publications

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“…One of the main goals of this paper is to make a further sept in determining the 'right size' of D n (Q, v). The following main result of this paper extends the lower bound given by (12) to the full range of v and to arbitrary degrees n.…”

Section: Introductionsupporting

confidence: 55%

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Integral polynomials with small discriminants and resultants

Beresnevich

Bernik

Götze

2016

Advances in Mathematics

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Let n ∈ N be fixed, Q > 1 be a real parameter and P n (Q) denote the set of polynomials over Z of degree n and height at most Q. In this paper we investigate the following counting problems regarding polynomials with small discriminant D(P ) and pairs of polynomials with small resultant R(P 1 , P 2 ):(i) given 0 ≤ v ≤ n − 1 and a sufficiently large Q, estimate the number of polynomials(ii) given 0 ≤ w ≤ n and a sufficiently large Q, estimate the number of pairs of polynomials P 1 , P 2 ∈ P n (Q) such that 0 < |R(P 1 , P 2 )| ≤ Q 2n−2w .Our main results provide lower bounds within the context of the above problems. We believe that these bounds are best possible as they correspond to the solutions of naturally arising linear optimisation problems. Using a counting result for the number of rational points near planar curves due to R. C. Vaughan and S. Velani we also obtain the complementary optimal upper bound regarding the discriminants of quadratic polynomials.

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Section: Introductionsupporting

confidence: 55%

“…More precisely, their formulation requires that the dicriminant/resultant is divisible by large powers of given prime numbers, see [6] for an overview. In recent years there has been substantial activity in attempting to resolve Problems 1 and 2, see [4,5,11,23,24,27], as well as their p-adic versions, see [12,15] and also [6].…”

Section: Introductionmentioning

confidence: 99%

Integral polynomials with small discriminants and resultants

Beresnevich

Bernik

Götze

2016

Advances in Mathematics

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“…The behavior of D(P ) is also closely related to the problem of the separation of conjugate algebraic numbers, which has been recently studied in some depth in [3,8,9,14]. Recently lower bounds for the number of integral polynomials with given heights and discriminants (or a discriminant divisible by a large prime power) have been obtained in [6,7]. Staying strictly within polynomials of degree 3, we obtain the asymptotic formula for the number of integral polynomials with bounded discriminants (Theorem 1 below).…”

Section: Introduction and Resultsmentioning

confidence: 99%

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

Kaliada

Götze²,

Kukso

2014

Lith Math J

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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 integer polynomials P such that H(P ) ≤ Q and |D(P )| ≤ X. We obtain the asymptotic formula of N (Q, X) for Q 14/5 ≪ X ≪ Q 4 and as Q → ∞. Using this result, for 0 ≤ η ≤ 0.9 we prove that H(P )≤Q 1≤|D(P )|≪Q 4−η |D(P )| −1/2 ≍ Q 2− η 3 for all sufficiently large Q, where the sum is taken over irreducible polynomials. This improves upon a result of Davenport who dealt with the case η = 0. We also consider an application of the main theorem to some outstanding problems of transcendental number theory.

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“…Also there are papers concerning this problem in similar settings with p-adic norm (instead of usual absolute value) used to bound discriminant values in (1) (see [18], [6], [4]). See also the survey [2].…”

Section: 3mentioning

confidence: 99%

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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On the divisibility of the discriminant of an integral polynomial by prime powers

Cited by 10 publications

References 7 publications

Integral polynomials with small discriminants and resultants

Integral polynomials with small discriminants and resultants

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

On the Distribution of Polynomial Discriminants: Totally Real Case

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