Distribution of complex algebraic numbers
type: Item
instance of: scholarly article
title: Distribution of complex algebraic numbers
zbMATH Open document ID: 1350.11087
author: Friedrich Götze
author: Dmitry Zaporozhets
author: Dzianis Kaliada
published in: Proceedings of the American Mathematical Society
publication date: 2016-11-14T00:00:00Z
full work available at URL: https://arxiv.org/abs/1410.3623
Mathematics Subject Classification ID: 11N45
Mathematics Subject Classification ID: 11J83
zbMATH DE Number: 6652844
zbMATH Keywords: distribution
zbMATH Keywords: algebraic numbers
zbMATH Keywords: polynomials with rational coefficients
MaRDI profile type: Publication
OpenAlex ID: W1850847674
arXiv ID: 1410.3623
cites work: On approximation of real numbers by real algebraic numbers
cites work: The distribution of close conjugate algebraic numbers
cites work: Q4515885
cites work: Distance between conjugate algebraic numbers in clusters
cites work: ON THE DISTANCE BETWEEN ROOTS OF INTEGER POLYNOMIALS
cites work: POLYNOMIAL ROOT SEPARATION
cites work: Q4829657
cites work: Root separation for irreducible integer polynomials
cites work: Root separation for reducible integer polynomials
cites work: The distribution of values of Mahler's measure
cites work: On a Principle of Lipschitz
cites work: On the distribution of roots of polynomials
cites work: Q3023596
cites work: О способе равномерного упорядочения вещественных алгебраических чисел
cites work: Discriminant and root separation of integral polynomials
cites work: On Distribution of Zeros of Random Polynomials in Complex Plane
cites work: Q3309989
cites work: Counting algebraic numbers with large height II
cites work: Counting Algebraic Numbers with Large Height I
cites work: Distribution of algebraic numbers
cites work: Q4137242
cites work: From explicit estimates for primes to explicit estimates for the Möbius function
cites work: Q5328476
cites work: Ergodic theory on \(SL(n)\), diophantine approximations and anomalies in the lattice point problem
cites work: Partial sums of the Möbius function
cites work: Die Seltenheit der reduziblen Gleichungen und der Gleichungen mit Affekt
DOI: 10.1090/PROC/13208
review text: In the paper under review, the authors give a formula for the number of algebraic numbers of height \(\leq Q\) and degree \(n\) in a bounded complex region \(\Omega\). It is assumed that \(\Omega\) does not intersect the real axis and that its boundary consists of a finite number of algebraic curves. Denoting the above number by \(\Psi_n(Q,\Omega)\), their formula is NEWLINE\[NEWLINE \Psi_n(Q,\Omega)=\frac{Q^{n+1}}{\zeta(n+1)} \int_{\Omega} \psi(z) \nu(dz)+O(Q^n), NEWLINE\]NEWLINE where \(\nu\) is the Lebesgue measure in the complex plane. The limit density \(\psi(z)\) is given a by a complicated function involving an integral over an \(n-1\) dimensional real region. The implicit constant in the big-O notation depends on \(n\), the number of algebraic curves which form the boundary of \(\Omega\) and their degrees. NEWLINENEWLINENEWLINEThe authors also give some estimates for the function \(\psi\) both near the real axis and away from it in the region \(|z|\geq 1\). The proof of the main result uses known bounds on the number of reducible polynomials of height \(\leq Q\) and degree \(n\), as well as known estimates for the number of integer points in the region \(tA\) as a function of the parameter \(t\) as \(t\to\infty\), where \(A\subset {\mathbb R}^d\) has boundary consisting of finitely many algebraic surfaces only.
reviewed by: Florian Luca
description: scientific article; zbMATH DE number 6652844