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

##### MSC:
 11G50 Heights 11J99 Diophantine approximation, transcendental number theory
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##### References:
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