An elementary inequality about the Mahler measure

Stulov, Konstantin; Yang, Rongwei

doi:10.2140/involve.2013.6.393

Download this article
	For screen
	For printing

Recent Issues

The Journal

About the journal

Ethics and policies

Peer-review process

Submission guidelines

ISSN 1944-4184 (online)

ISSN 1944-4176 (print)

Author index

To appear

Other MSP journals

An elementary inequality about the Mahler measure

Konstantin Stulov and Rongwei Yang

Vol. 6 (2013), No. 4, 393–397

DOI: 10.2140/involve.2013.6.393

Abstract

Let $p (z)$ be a degree $n$ polynomial with zeros $z_{j}, j = 1, 2, \dots, n$ . The total distance from the zeros of $p$ to the unit circle is defined as $td (p) = \sum_{j = 1}^{n} | | z_{j} | - 1 |$ . We show that up to scalar multiples, $td (p)$ sits between $M (p) - 1$ and $m (p)$ . This leads to an equivalent statement of Lehmer’s problem in terms of $td (p)$ . The proof is elementary.

Keywords

Mahler measure, total distance

Mathematical Subject Classification 2010

Primary: 11CXX

Milestones

Received: 9 July 2012

Revised: 12 February 2013

Accepted: 16 February 2013

Published: 8 October 2013

Communicated by Andrew Granville

Authors

Konstantin Stulov
Institute for Computational and Mathematical Engineering Stanford University Stanford, NY 94305 United States

Rongwei Yang
Department of Mathematics and Statistics University of Albany State University of New York Albany, NY 12047 United States

Vol. 6, No. 4, 2013