Orbits of algebraic numbers with low heights

Dresden, Gregory

doi:10.1090/S0025-5718-98-00963-6

Orbits of algebraic numbers with low heights
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by Gregory P. Dresden PDF

Math. Comp. 67 (1998), 815-820 Request permission

Abstract:

We prove that the two smallest values of $h(\alpha ) + h(\frac {1}{1-\alpha }) + h(1 - \frac {1}{\alpha })$ are $0$ and $0.4218\dots$, for $\alpha$ any algebraic integer.

References

E. Bombieri, A van der Poorten, and J. Vaaler, Effective measures of irrationality for cubic extensions of number fields, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 23 (1996), 211–248.
David W. Boyd, Speculations concerning the range of Mahler’s measure, Canad. Math. Bull. 24 (1981), no. 4, 453–469. MR 644535, DOI 10.4153/CMB-1981-069-5
E. Dobrowolski, On a question of Lehmer and the number of irreducible factors of a polynomial, Acta Arith. 34 (1979), no. 4, 391–401. MR 543210, DOI 10.4064/aa-34-4-391-401
D. H. Lehmer, Factorization of certain cyclotomic funcions, Ann. of Math. 34 (1933), 461–479.
Christopher G. Pinner and Jeffrey D. Vaaler, The number of irreducible factors of a polynomial. I, Trans. Amer. Math. Soc. 339 (1993), no. 2, 809–834. MR 1150018, DOI 10.1090/S0002-9947-1993-1150018-X
Ulrich Rausch, On a theorem of Dobrowolski about the product of conjugate numbers, Colloq. Math. 50 (1985), no. 1, 137–142. MR 818097, DOI 10.4064/cm-50-1-137-142
D. Zagier, Algebraic numbers close to both $0$ and $1$, Math. Comp. 61 (1993), no. 203, 485–491. MR 1197513, DOI 10.1090/S0025-5718-1993-1197513-9
Shouwu Zhang, Positive line bundles on arithmetic surfaces, Ann. of Math. (2) 136 (1992), no. 3, 569–587. MR 1189866, DOI 10.2307/2946601