Distribution of algebraic numbers

作者：Pritsker Igor E^*

来源：Journal fur die Reine und Angewandte Mathematik, 2011, 657: 57-80.

DOI：10.1515/CRELLE.2011.049

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 lim sup (n ->infinity) vertical bar A(n)vertical bar <= 1 - root e/2 We show that A(n) -> 0, and estimate the rate of convergence by generalizing the Erdos-Turan theorem on and estimate the rate of convergence by generalizing the Erdos-Turan 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 lim inf (n -> infinity) A(n) 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.

• 出版日期2011-8