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Zbl 1016.11026
Amoroso, Francesco; Zannier, Umberto
A relative Dobrowolski lower bound over abelian extensions. (English)
[J] Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 29, No.3, 711-727 (2000). ISSN 0391-173X

Let $K$ be a number field, $\alpha\ne 0$ an algebraic number which is not a root of unity. Lehmer's problem [{\it D. H. Lehmer}, Ann. Math. (2) 34, 461-479 (1933; Zbl 0007.19904)] consists in asking for an absolute constant $c_0>0$ such that $h(\alpha)\ge\frac{c_0}{[\Bbb{Q}(\alpha):\Bbb{Q}]}$, where $h(\alpha)$ denotes the absolute logarithmic height of $\alpha$. This problem remains still open and the best unconditional bound already obtained is due to {\it E. Dobrowolski} [Acta Arith. 34, 391-401 (1979; Zbl 0416.12001)], $h(\alpha)\ge\frac{c_1}d (\frac{\log(3d)}{\log\log(3d)})^{-3}$, where $D=[\Bbb{Q}(\alpha):\Bbb{Q}]$ and $c_1>0$ is an absolute constant. In some special cases, not only Lehmer's inequality is true but also sharper bounds are obtained. Suppose $\Bbb{Q}(\alpha)/\Bbb{Q}$ is an abelian extension. Then the first author and {\it R. Dvornicich} proved in [J. Number Theory 80, 260-272 (2000; Zbl 0973.11092)] that $h(\alpha)\ge\frac{\log 5}{12}$. In fact, this result is a special case of a more general one due to {\it A. Schinzel} [Acta Arith. 24, 385-399 (1973; Zbl 0275.12004), Addendum ibid. 26, 329-361 (1973; Zbl 0312.12001)], but with the extra hypothesis that $|\alpha|\ne 1$. The main goal of the paper is to generalize both results. \par More precisely, let $K$ be a number field and $L$ an abelian extension of $K$. Then for every nonzero algebraic number $\alpha$ which is not a root of unity, $$h(\alpha)\ge\frac{c_2(K)}d\left(\frac{\log(2d)}{\log\log(5d)}\right)^{-13},$$ where $d=[L(\alpha):L]$ and $c_2(K)>0$ is a constant depending on $K$.\par Recently a result due to the second author and E. Bombieri showed that if $K$ is a number field and $L$ is the compositum of all extensions of $K$ of degree at most $d$, then given $T>0$ the number of elements of $L$ of height at most $T$ is finite.\par Dobrowolski's result can be phrased in terms of Mahler measure as follows. Let $F\in\Bbb{Z}[x]$ and suppose $\alpha$ is not a root of $F$, then $\log M(F)=\deg(F)h(\alpha)$, where $M(F)$ denotes the Mahler measure of $F$. The quoted result is expressed as $\log M(F)\ge c_1(\frac{\log(3d)}{\log\log(3d)})^{-3}$, where $d=\deg(F)$ and we assume that $F$ is not a cyclotomic polynomial. The first author and {\it S. David} extended this result to polynomials in $n$ variables in [Acta Arith. 92, 339-366 (2000; Zbl 0948.11025)] obtaining $$\log M(F)\ge\frac 1{c_3(n+1)^{1+4/n}n^2}\left(\frac{\log((n+1)d)}{\log((n+1)\log((n+1)d))}\right)^{-3},$$ where $c_3>0$ is an absolute constant and $d=\deg(f)$. In this paper, as a consequence of the first result cited above and a density result of F. Amoroso and S. David (loc. cit.), it is proved that if $F\in\Bbb{Z}[x_1,\cdots,x_n]$ is irreducible and not an extended cyclotomic polynomial and $d=\min_{j=1,\cdots,n}\deg_{x_j}(F)\ge 1$, then $$\log M(F)\ge c_2(\Bbb{Q})\left(\frac{\log(2d)}{\log\log(5d)}\right)^{-13}.$$ This estimate is stronger than the one in [F. Amoroso and S. David, loc. cit.] if at least one of the partial degrees of $F$ is small.
[Amílcar Pacheco (Rio de Janeiro)]

MSC 2000:: *11G50 Heights
11J99 Diophantine approximation

Keywords: heights; absolute logarithmic height; Lehmer problem; lower bound; cyclotomic polynomials

Citations: Zbl 0007.19904; Zbl 0416.12001; Zbl 0973.11092; Zbl 0275.12004; Zbl 0312.12001; Zbl 0948.11025

Cited in: Zbl 1205.11079 Zbl 1204.11100 Zbl 1091.11020 Zbl 1114.11058

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