A relative Dobrowolski lower bound over abelian extensions. (English) Zbl 1016.11026

Amoroso, Francesco; Zannier, Umberto

Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 29, No. 3, 711-727 (2000).

Let \(K\) be a number field, \(\alpha\neq 0\) an algebraic number which is not a root of unity. Lehmer’s problem [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 \(h(\alpha)\geq\frac{c_0}{[\mathbb{Q}(\alpha):\mathbb{Q}]}\), 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 E. Dobrowolski [Acta Arith. 34, 391-401 (1979; Zbl 0416.12001)], \(h(\alpha)\geq\frac{c_1}d (\frac{\log(3d)}{\log\log(3d)})^{-3}\), where \(D=[\mathbb{Q}(\alpha):\mathbb{Q}]\) 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 \(\mathbb{Q}(\alpha)/\mathbb{Q}\) is an abelian extension. Then the first author and R. Dvornicich proved in [J. Number Theory 80, 260-272 (2000; Zbl 0973.11092)] that \(h(\alpha)\geq\frac{\log 5}{12}\). In fact, this result is a special case of a more general one due to 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 \(|\alpha|\neq 1\). The main goal of the paper is to generalize both results.
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, \[ h(\alpha)\geq\frac{c_2(K)}d\left(\frac{\log(2d)}{\log\log(5d)}\right)^{-13}, \] where \(d=[L(\alpha):L]\) and \(c_2(K)>0\) is a constant depending on \(K\).
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.
Dobrowolski’s result can be phrased in terms of Mahler measure as follows. Let \(F\in\mathbb{Z}[x]\) 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 \(\log M(F)\geq c_1(\frac{\log(3d)}{\log\log(3d)})^{-3}\), where \(d=\deg(F)\) and we assume that \(F\) is not a cyclotomic polynomial. The first author and S. David extended this result to polynomials in \(n\) variables in [Acta Arith. 92, 339-366 (2000; Zbl 0948.11025)] obtaining \[ \log M(F)\geq\frac 1{c_3(n+1)^{1+4/n}n^2}\left(\frac{\log((n+1)d)}{\log((n+1)\log((n+1)d))}\right)^{-3}, \] 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 \(F\in\mathbb{Z}[x_1,\cdots,x_n]\) is irreducible and not an extended cyclotomic polynomial and \(d=\min_{j=1,\cdots,n}\deg_{x_j}(F)\geq 1\), then \[ \log M(F)\geq c_2(\mathbb{Q})\left(\frac{\log(2d)}{\log\log(5d)}\right)^{-13}. \] 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.

Reviewer: Amílcar Pacheco (Rio de Janeiro)

MSC:

11G50	Heights
11J99	Diophantine approximation, transcendental number theory

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

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