Distribution of algebraic numbers
Abstract
Schur studied limits of the arithmetic means An of zeros for polynomials of degree n with integer coefficients and simple zeros in the closed unit disk. If the leading coefficients are bounded, Schur proved that 
. We show that An → 0, and estimate the rate of convergence by generalizing the Erdős–Turán theorem on the distribution of zeros. As an application, we show that integer polynomials have some unexpected restrictions of growth on the unit disk.
Schur also studied problems on means of algebraic numbers on the real line. When all conjugate algebraic numbers are positive, the problem of finding the sharp lower bound for 
was developed further by Siegel and others. We provide a solution of this problem for algebraic numbers equidistributed in subsets of the real line.
Potential theoretic methods allow us to consider the distribution of algebraic numbers in or near general sets in the complex plane. We introduce the generalized Mahler measure, and use it to characterize asymptotic equidistribution of algebraic numbers in arbitrary compact sets of capacity one. The quantitative aspects of this equidistribution are also analyzed in terms of the generalized Mahler measure.
© Walter de Gruyter Berlin · New York 2011
Articles in the same Issue
- Coefficients and non-triviality of the Jones polynomial
- Distribution of algebraic numbers
- Some local-global non-vanishing results of theta lifts for symplectic-orthogonal dual pairs
- Characterizing quaternion rings over an arbitrary base
- Transcendence in positive characteristic and special values of hypergeometric functions
- Factoring 3-fold flips and divisorial contractions to curves
- Existence of permanent and breaking waves for the periodic Degasperis–Procesi equation with linear dispersion
- -actions on UHF algebras of infinite type
Articles in the same Issue
- Coefficients and non-triviality of the Jones polynomial
- Distribution of algebraic numbers
- Some local-global non-vanishing results of theta lifts for symplectic-orthogonal dual pairs
- Characterizing quaternion rings over an arbitrary base
- Transcendence in positive characteristic and special values of hypergeometric functions
- Factoring 3-fold flips and divisorial contractions to curves
- Existence of permanent and breaking waves for the periodic Degasperis–Procesi equation with linear dispersion
- -actions on UHF algebras of infinite type
