arvind@utcsri.UUCP (07/28/87)
Date: 28 Jul 1987 09:59:19-EDT (Tuesday)
From: "Victor S. Miller" <VICTOR@yktvmz.bitnet>
Subject: Lehmer's conjecture
Since the last response, Andrew Odlyzko sent me one, reference, and
I found a few more:
The references to Cantor and Straus and Dobrowolski that I gave
previously, prove that there exist a constant $c$, such that given
$\epsilon > 0$ there is a $D(\epsilon)$ so that for all algebraic
integers $\alpha$ of degree $>D(\epsilon)$ , not a root of unity, we have
$M(\alpha) > 1 + c(\log \log d / \log d)^3$
The values of $c$ given are $c=1$ (Dobrowolski), $c=2$ (Cantor and Straus)
and $c=9/4$ (Louboutin [1]). However, Smyth [2] has proven that there
exits $C>1$ independent of the degree, such that if $\alpha$ is not
reciprocal (\alpha reciprocal means that \alpha and 1/\alpha are
conjugate) then $M(\alpha)>C$ when $\alpha$ is not a root of unity.
Meanwhile papers by Smyth [3,4], Schinzel [5], and finally Langenvin [6,7]
have proven results of the following sort:
Given an open set $V$ of the complex plane containing at least one
point on the unit circle, there is an effectively computable constant
$C(V)>1$ such that if $\alpha$ and all its conjugates are not contained
in $V$ then $M(\alpha)>C(V)^d$ where $d=\deg \alpha$. The Smyth and
Schinzel papers deal with the cases where $V$ is either the complement
of the real axis or the set of positive reals. Langevin deals with the
general case.
Victor S. Miller -- IBM Research
[1] Roland Louboutin, "Sur la mesure de Mahler d'un nombre algebrique",
C.R. Acad. Sc. Paris, T. 296, pp 707-708
[2] C.J. Smyth, "On the product of conjugates outside the unit circle
of an algebraic integer", Bull. London Math Soc., 3(1971), pp 169-175.
[3] C.J. Smyth, "On the measure of totally real algebraic integers",
J. Austral. Math. Soc. (Series A), 30(1980), pp 137-149.
[4] C.J. Smyth, "On the measure of totally real algebraic integers, II",
Math. Comp. 37(1981), pp 205-208.
[5]A. Schinzel, "On the product of the conjugates outside the unit
circle of an algebraic number", Acta Arithmetica, 24(1973), pp. 385-399.
[6] Michel Langevin, "Methode de Fekete-Szego et probleme de Lehmer",
C.R. Acad. Sc. Paris, t. 301 (1985), pp 463-466.
[7] Michel Langevin, "Minorations de la maison et de la mesure de Mahler
de certains entiers algebriques", C.R. Acad. Sc. Paris, t. 303(1986),
pp. 523-526.