English
 
Help Privacy Policy Disclaimer
  Advanced SearchBrowse

Item

ITEM ACTIONSEXPORT

Released

Journal Article

Algebraic numbers close to both 0 and 1

MPS-Authors
/persons/resource/persons236497

Zagier,  Don
Max Planck Institute for Mathematics, Max Planck Society;

External Resource
No external resources are shared
Fulltext (public)
There are no public fulltexts stored in PuRe
Supplementary Material (public)
There is no public supplementary material available
Citation

Zagier, D. (1993). Algebraic numbers close to both 0 and 1. Mathematics of Computation, 61(203), 485-491. doi:10.2307/2152970.


Cite as: http://hdl.handle.net/21.11116/0000-0004-3903-9
Abstract
Let α be a number algebraic over the rationals and let H(α) denote the absolute logarithmic height of α, which can be defined as H(α)= \\log M(f)\\sp1/n, where α is a root of the irreducible polynomial f(x) with rational coefficients and degree n, and where M(f) denotes the Mahler measure of f. The author gives an elementary proof of a sharp version of a remarkable inequality of Shouwu Zhang. He shows that, for all α\\ne 0,1, (1\\pm\\sqrt-3)/2, the following inequality holds: H(α)+ H(1-α)≥q 1\\over 2 \\log 1+\\sqrt5\\over 2= 0.2406059\\dots, with equality if and only if α or 1-α is a primitive 10th root of unity. He also proves a sharp projective version of this inequality for the curve x+y+z=0 and gives an outline of how to prove similar results for other curves.