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The Lehmer constants of an annulus. (English) Zbl 1030.11057
The Lehmer constant $$L(V)$$ of an open set $$V$$ in the complex plane is defined [M. Langevin, C. R. Acad. Sci., Paris, Sér I. 301, 463-466 (1985; Zbl 0585.12013)] by $$L(V)=\inf M(\alpha)^{1/\deg(\alpha)}$$, the infimum taken over all non-zero non-cyclotomic algebraic integers $$\alpha$$ lying with all conjugates outside $$V$$, and $$M(\alpha)$$ denotes the Mahler measure of $$\alpha$$. The authors compute $$L(V)$$ in the case when $$V$$ is an annulus centered at the origin and prove that for any open set $$V$$ the inequality $$L(V)>1$$ implies that $$V$$ has a non-empty intersection with the unit circle (the converse implication has been earlier established by Langevin (loc.cit.)) Moreover they consider a variant of Lehmer’s constant, namely $$L_\infty(V)=\lim_{d\to\infty}\inf M(\alpha)^{1/\deg(\alpha)}$$, the infimum being taken over all $$\alpha$$ of degree $$\geq d$$ lying with their conjugates outside $$V$$, and obtain similar results.

##### MSC:
 11R04 Algebraic numbers; rings of algebraic integers 11R06 PV-numbers and generalizations; other special algebraic numbers; Mahler measure 11R09 Polynomials (irreducibility, etc.) 12D10 Polynomials in real and complex fields: location of zeros (algebraic theorems)
##### Keywords:
Lehmer constant; Mahler measure; polynomial zeros
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##### References:
 [1] Beukers, F., Smyth, C.J., 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). · Zbl 1029.11009 [2] Dubickas, A., On the distribution of roots of polynomials in sectors. I. Liet. Matem. Rink.38 (1998), 34-58. · Zbl 0920.12001 [3] Dubickas, A., Smyth, C.J., 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. · Zbl 0997.11087 [4] Langevin, M., Méthode de Fekete - Szegö et problème de Lehmer. C. R. Acad. Sci. Paris Sér. I Math.301 (1985), 463-466. · Zbl 0585.12013 [5] Langevin, M., 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. · Zbl 0604.12001 [6] Langevin, M., 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. · Zbl 0678.12002 [7] Lehmer, D.H., Factorisation of certain cyclotomic functions. Ann. of Math. (2) 34 (1933), 461-479. · JFM 59.0933.03 [8] Lehmer, D.H., Review of [La3]. Math. Rev.89j:11025. [9] Mignotte, M., Sur un théorème de M. Langevin. Acta Arith.54 (1989), 81-86. · Zbl 0641.12003 [10] Rhin, G., Smyth, C.J., On the absolute Mahler measure of polynomials having all zeros in a sector. Math. Comp.65 (1995), 295-304. · Zbl 0820.11064 [11] Schinzel, A., 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. · Zbl 0312.12001 [12] Smyth, C.J., On the measure of totally real algebraic integers. J. Austral. Math. Soc. Ser. A30 (1980), 137-149. · Zbl 0457.12001
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