Involve: A Journal of Mathematics

  • Involve
  • Volume 6, Number 4 (2013), 393-397.

An elementary inequality about the Mahler measure

Konstantin Stulov and Rongwei Yang

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Abstract

Let p(z) be a degree n polynomial with zeros zj,j=1,2,,n. The total distance from the zeros of p to the unit circle is defined as td(p)=j=1n||zj|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.

Article information

Source
Involve, Volume 6, Number 4 (2013), 393-397.

Dates
Received: 9 July 2012
Revised: 12 February 2013
Accepted: 16 February 2013
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.involve/1513733607

Digital Object Identifier
doi:10.2140/involve.2013.6.393

Mathematical Reviews number (MathSciNet)
MR3115974

Zentralblatt MATH identifier
1327.11077

Subjects
Primary: 11CXX

Keywords
Mahler measure total distance

Citation

Stulov, Konstantin; Yang, Rongwei. An elementary inequality about the Mahler measure. Involve 6 (2013), no. 4, 393--397. doi:10.2140/involve.2013.6.393. https://projecteuclid.org/euclid.involve/1513733607


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References

  • G. Everest and T. Ward, Heights of polynomials and entropy in algebraic dynamics, Springer, London, 1999.
  • J. B. Garnett, Bounded analytic functions, 1st ed., Graduate Texts in Mathematics 236, Springer, New York, 2007.
  • C. Smyth, “The Mahler measure of algebraic numbers: a survey”, pp. 322–349 in Number theory and polynomials, London Math. Soc. Lecture Note Ser. 352, Cambridge Univ. Press, 2008.