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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:
[1] F. Amoroso - S. David , Le problème de Lehmer en dimension supérieure , J. reine angew. Math . 513 ( 1999 ), 145 - 179 . MR 1713323 | Zbl 1011.11045 · Zbl 1011.11045 · doi:10.1515/crll.1999.058
[2] F. Amoroso - S. David , Minoration de la hauteur normalisée des hypersurfaces , Acta Arith . 92 ( 2000 ), no. 4 , 340 - 366 . Article | MR 1760242 | Zbl 0948.11025 · Zbl 0948.11025 · eudml:207392
[3] F. Amoroso - R. Dvornicich , A lower bound for the height in Abelian extensions , J. Number Theory 80 ( 2000 ), no. 2 , 260 - 272 . MR 1740514 | Zbl 0973.11092 · Zbl 0973.11092 · doi:10.1006/jnth.1999.2451
[4] D. Boyd , Kronecker’s theorem and Lehmer’s problem for polynomials in several variables , J. Number Theory 13 ( 1980 ), 116 - 121 . MR 602452 | Zbl 0447.12003 · Zbl 0447.12003 · doi:10.1016/0022-314X(81)90033-0
[5] P. Philippon - S. David , Minorations des hauteurs normalisées des sous-variétés des tores , Ann. Scuola Norm. Sup. Pisa a. Sci ( 4 ) 28 ( 1999 ), 489 - 543 . Numdam | MR 1736526 | Zbl 1002.11055 · Zbl 1002.11055 · numdam:ASNSP_1999_4_28_3_489_0 · eudml:84386
[6] E. Dobrowolski , On a question of Lehmer and the number of irreducible factors of a polynomial , Acta Arith . 34 ( 1979 ), 391 - 401 . Article | MR 543210 | Zbl 0416.12001 · Zbl 0416.12001 · eudml:205618
[7] M. Laurent , Equations diophantiennes exponentielles , Invent. Math. 78 ( 1984 ), 299 - 327 . MR 767195 | Zbl 0554.10009 · Zbl 0554.10009 · doi:10.1007/BF01388597 · eudml:143175
[8] W. Lawton , A generalization of a theorem of Kronecker , J. of the Science Faculty of Chiangmai University ( Thailand ) 4 ( 1977 ), 15 - 23 . Zbl 0447.12004 · Zbl 0447.12004
[9] D.H. Lehmer , Factorization of certain cyclotomic functions , Ann. of Math . 34 ( 1933 ), 461 - 479 . MR 1503118 | Zbl 0007.19904 | JFM 59.0933.03 · Zbl 0007.19904 · doi:10.2307/1968172 · www.emis.de
[10] W. Narkiewicz , ”Elementary and analytic theory of algebraic numbers” , Second edition, Springer-Verlag, Berlin, PWN-Polish Scientific Publishers , Warsaw , 1990 . MR 1055830 | Zbl 0717.11045 · Zbl 0717.11045
[11] N. Northcott , An inequality in the theory of arithmetic on algebraic varieties , Proc. Camb. Philos. Soc. 45 ( 1949 ), 502 - 509 . MR 33094 | Zbl 0035.30701 · Zbl 0035.30701
[12] U. Rausch , On a theorem of Dobrowolski about the product of conjugate numbers , Colloq. Math. 50 ( 1985 ), no. 1 , 137 - 142 . MR 818097 | Zbl 0579.12001 · Zbl 0579.12001
[13] D. Roy - J. Thunder , An absolute Siegel’s lemma , J. reine angew. Math . 476 ( 1996 ), 1 - 12 . MR 1401695 | Zbl 0860.11036 · Zbl 0860.11036 · crelle:GDZPPN002213745 · eudml:153828
[14] A. Schinzel , On the product of the conjugates outside the unit circle of an algebraic number , Acta Arith . 24 ( 1973 ), 385 - 399 . Addendum, ibid . 26 ( 1973 ), 329 - 361 . Article | MR 360515 | Zbl 0275.12004 · Zbl 0275.12004 · eudml:205236
[15] W.M. Schmidt , ” Diophantine approximations and Diophantine equations ”, Lecture Notes in Mathematics 1467 , Springer-Verlag , Berlin , 1991 . MR 1176315 | Zbl 0754.11020 · Zbl 0754.11020 · doi:10.1007/BFb0098246
[16] C.J. Smyth , A Kronecker-type theorem for complex polynomials in several variables , Canad. Math. Bull. 24 ( 1981 ), 447 - 452 . Errata, ibid . 25 ( 1982 ), 504 . MR 644534 | Zbl 0475.12002 · Zbl 0475.12002 · doi:10.4153/CMB-1981-068-8
[17] L.C. Washington , ” Introduction to Cyclotomic Fields ”, Springer-Verlag, New York , 1982 . MR 718674 | Zbl 0484.12001 · Zbl 0484.12001
[18] S. Zhang , Positive line bundles on arithmetic surfaces , Ann. of Math . 136 ( 1992 ), 569 - 587 . MR 1189866 | Zbl 0788.14017 · Zbl 0788.14017 · doi:10.2307/2946601
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