Distribution of Algebraic Numbers and Metric Theory of Diophantine Approximation

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

doi:10.1007/978-3-642-36068-8_2

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Introduction and Main Results1

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2019

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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: Introduction and Main Resultsmentioning

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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Section: Introduction and Main Resultsmentioning

confidence: 99%

On the Distribution of Polynomial Discriminants: Totally Real Case

Koleda

2019

Lith Math J

View full text Add to dashboard Cite

show abstract

Distribution of Algebraic Numbers and Metric Theory of Diophantine Approximation

Cited by 1 publication

References 32 publications

On the Distribution of Polynomial Discriminants: Totally Real Case

On the Distribution of Polynomial Discriminants: Totally Real Case

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