The Lehmer constants of an annulus

Artūras Dubickas; Chris J. Smyth

Journal de théorie des nombres de Bordeaux (2001)

  • Volume: 13, Issue: 2, page 413-420
  • ISSN: 1246-7405

Abstract

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Let M ( α ) be the Mahler measure of an algebraic number α , and V be an open subset of ℂ . Then its Lehmer constant L ( V ) is inf M ( α ) 1 / deg ( α ) , the infimum being over all non-zero non-cyclotomic α lying with its conjugates outside V . We evaluate L ( V ) when V is any annulus centered at 0 . We do the same for a variant of L ( V ) , which we call the transfinite Lehmer constant L ∞ ( V ) .Also, we prove the converse to Langevin’s Theorem, which states that L ( V ) > 1 if V contains a point of modulus 1 . We prove the corresponding result for L ∞ ( V ) .

How to cite

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Dubickas, Artūras, and Smyth, Chris J.. "The Lehmer constants of an annulus." Journal de théorie des nombres de Bordeaux 13.2 (2001): 413-420. <http://eudml.org/doc/248715>.

@article{Dubickas2001,
abstract = {Let $M (\alpha )$ be the Mahler measure of an algebraic number $\alpha $, and $V$ be an open subset of $\mathbb \{C\}$. Then its Lehmer constant$L (V)$ is inf $M (\alpha )^\{1/ \deg (\alpha )\}$, the infimum being over all non-zero non-cyclotomic $\alpha $ lying with its conjugates outside $V$. We evaluate $L(V)$ when $V$ is any annulus centered at $0$. We do the same for a variant of $L (V)$, which we call the transfinite Lehmer constant $L_\infty (V)$.Also, we prove the converse to Langevin’s Theorem, which states that $L (V) &gt; 1$ if $V$ contains a point of modulus $1$. We prove the corresponding result for $L_\infty (V)$.},
author = {Dubickas, Artūras, Smyth, Chris J.},
journal = {Journal de théorie des nombres de Bordeaux},
keywords = {Lehmer constant; Mahler measure; polynomial zeros},
language = {eng},
number = {2},
pages = {413-420},
publisher = {Université Bordeaux I},
title = {The Lehmer constants of an annulus},
url = {http://eudml.org/doc/248715},
volume = {13},
year = {2001},
}

TY - JOUR
AU - Dubickas, Artūras
AU - Smyth, Chris J.
TI - The Lehmer constants of an annulus
JO - Journal de théorie des nombres de Bordeaux
PY - 2001
PB - Université Bordeaux I
VL - 13
IS - 2
SP - 413
EP - 420
AB - Let $M (\alpha )$ be the Mahler measure of an algebraic number $\alpha $, and $V$ be an open subset of $\mathbb {C}$. Then its Lehmer constant$L (V)$ is inf $M (\alpha )^{1/ \deg (\alpha )}$, the infimum being over all non-zero non-cyclotomic $\alpha $ lying with its conjugates outside $V$. We evaluate $L(V)$ when $V$ is any annulus centered at $0$. We do the same for a variant of $L (V)$, which we call the transfinite Lehmer constant $L_\infty (V)$.Also, we prove the converse to Langevin’s Theorem, which states that $L (V) &gt; 1$ if $V$ contains a point of modulus $1$. We prove the corresponding result for $L_\infty (V)$.
LA - eng
KW - Lehmer constant; Mahler measure; polynomial zeros
UR - http://eudml.org/doc/248715
ER -

References

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  1. [BS] F. Beukers, C.J. Smyth, Cyclotomic points on curves. Number Theory for the Millennium: Proc. Millennial Conf. Number Theory (B. C. Berndt et al., eds.), Urbana, Illinois, May 21 - 26, 2000, A K Peters, Ltd., Natick, MA (to appear 2002). Zbl1029.11009MR1956219
  2. [D] A. Dubickas, On the distribution of roots of polynomials in sectors. I. Liet. Matem. Rink.38 (1998), 34-58. Zbl0920.12001MR1663816
  3. [DS] A. Dubickas, C.J. Smyth, On the Remak height, the Mahler measure, and conjugate sets of algebraic numbers lying on two circles. Proc. Edinburgh Math. Soc.44 (2001), 1-17. Zbl0997.11087MR1878724
  4. [La] M. Langevin, Méthode de Fekete - Szegö et problème de Lehmer. C. R. Acad. Sci. Paris Sér. I Math.301 (1985), 463-466. Zbl0585.12013MR812558
  5. [La2] M. Langevin, Minorations de la maison et de la mesure de Mahler de certains entiers algébriques. C. R. Acad. Sci. Paris Sèr. I Math.303 (1986), 523-526. Zbl0604.12001MR867930
  6. [La3] M. Langevin, Calculs explicites de constantes de Lehmer. Groupe de travail en théorie analytique et élémentaire des nombres, 1986-1987, 52-68, Publ. Math. Orsay, 88-01, Univ. Paris XI, Orsay, 1988. Zbl0678.12002MR950948
  7. [Le1] D.H. Lehmer, Factorisation of certain cyclotomic functions. Ann. of Math. (2) 34 (1933), 461-479. Zbl0007.19904MR1503118JFM59.0933.03
  8. [Le2] D.H. Lehmer, Review of [La3]. Math. Rev.89j:11025. 
  9. [M] M. Mignotte, Sur un théorème de M. Langevin. Acta Arith.54 (1989), 81-86. Zbl0641.12003MR1024420
  10. [RS] G. Rhin, C.J. Smyth, On the absolute Mahler measure of polynomials having all zeros in a sector. Math. Comp.65 (1995), 295-304. Zbl0820.11064MR1257579
  11. [Sc] A. Schinzel, On the product of the conjugates outside the unit circle of an algebraic number. Acta Arith.24 (1973), 385-399; Addendum 26 (1975), 329-331. Zbl0275.12004MR360515
  12. [Sm] C.J. Smyth, On the measure of totally real algebraic integers. J. Austral. Math. Soc. Ser. A30 (1980), 137-149. Zbl0457.12001MR607924

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