New estimates for the number of integer polynomials with given discriminants
type: Item
instance of: scholarly article
title: New estimates for the number of integer polynomials with given discriminants
zbMATH Open document ID: 1451.11083
author: Natalia V. Budarina
author: Hugh O'Donnell
author: V. I. Bernik
published in: Lithuanian Mathematical Journal
publication date: 2020-07-14T00:00:00Z
full work available at URL: https://arrow.tudublin.ie/cgi/viewcontent.cgi
review text: The present article deals with an upper bound for the number of polynomials \(P\in \mathcal P^{'} _4 (Q,v)\). Here \(\mathcal P^{'} _4 (Q,v)\) is the set of polynomials \(P\in \mathcal P_4 (Q,v)\) with distinct roots satisfying certain conditions. The authors present a new method of upper bounds for the number of integer polynomials of the fourth degree with a given discriminant. Connections between the present research and known results are described. A brief survey is devoted to the problem of counting polynomials with a small discriminant \(D(P)\) in the theory of Diophantine approximations. All auxiliary notions are recalled, and some statements which are useful for proving the main result are given.
reviewed by: Symon Serbenyuk
Mathematics Subject Classification ID: 11J83
Mathematics Subject Classification ID: 11J68
zbMATH DE Number: 7220249
zbMATH Keywords: Diophantine approximation
zbMATH Keywords: discriminant of polynomials
MaRDI profile type: Publication
OpenAlex ID: W2982284651
cites work: Rational points near manifolds and metric Diophantine approximation
cites work: Simultaneous approximations of zero by an integral polynomial its derivative, and small values of discriminants
cites work: Integral polynomials with small discriminants and resultants
cites work: Exact upper bounds for the number of the polynomials with given discriminants
cites work: Discriminants of polynomials in the Archimedean and non-Archimedean metrics
cites work: Lower bounds for the number of integral polynomials with given order of discriminants
cites work: On the divisibility of the discriminant of an integral polynomial by prime powers
cites work: HOW SMALL CAN POLYNOMIALS BE IN AN INTERVAL OF GIVEN LENGTH?
cites work: Применение размерностн Хаусдорфа в теории диофантовых приближений
cites work: О точном порядке приближения нуля значениями целочисленных многочленов
cites work: On the number of polynomials with small discriminants in the Euclidean and \(p\)-adic metrics
cites work: Q4829657
cites work: The asymptotic number of integral cubic polynomials with bounded heights and discriminants
cites work: Q5571008
cites work: Q5625295
Wikidata QID: Q126983393
DOI: 10.1007/S10986-019-09455-8
description: scientific article; zbMATH DE number 7220249