Two new points in the spectrum of the absolute Mahler measure of totally positive algebraic integers

Flammang, V.

doi:10.1090/S0025-5718-96-00664-3

Two new points in the spectrum of the absolute Mahler measure of totally positive algebraic integers
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Math. Comp. 65 (1996), 307-311 Request permission

Abstract:

For totally positive algebraic integers $\alpha \ne 0,1$ of degree $d(\alpha )$, we consider the set $\mathcal {L}$ of values of $M(\alpha )^{\frac 1{d(\alpha )}}=\Omega (\alpha )$, where $M(\alpha )$ is the Mahler measure of $\alpha$. C. J. Smyth has found the four smallest values of $\mathcal {L}$ and conjectured that the fifth point is $\Omega ((2\cos \frac {2\pi }{60})^2)$. We prove that this is so and, moreover, we give the sixth point of $\mathcal {L}$.

References

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Sur la longueur des entiers algébriques totalement positifs

C. J. Smyth, On the measure of totally real algebraic integers, J. Austral. Math. Soc. Ser. A 30 (1980/81), no. 2, 137–149. MR 607924, DOI 10.1017/S1446788700016426
C. J. Smyth, On the measure of totally real algebraic integers, J. Austral. Math. Soc. Ser. A 30 (1980/81), no. 2, 137–149. MR 607924, DOI 10.1017/S1446788700016426
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