Calculs explicites de constantes de Lehmer. (Explicit calculations of Lehmer’s constants). (French) Zbl 0678.12002

Langevin, Michel

Publ. Math. Orsay 88/01, 52-68 (1988).

The Mahler measure M(P) of a polynomial \(P(z)=a_ 0z^ d+...+a_ d\) with complex coefficients, and zeros \(x_ 1,...,x_ d\), is the product \(| a_ 0| \prod_{i}\max (1,| x_ i|).\)
If \(\alpha\) is an algebraic number with minimal polynomial P, then one defines \(M(\alpha):=M(P)\). - Using the concept of transfinite diameter, the author has shown that, if V is a neighbourhood of a point on the unit circle, there is a constant \(C(V)>1\) such that if \(\alpha\) is an algebraic integer of degree \( d\) which is not a root of unity and such that none of its conjugates lie in V, then \(M(\alpha)>C(V)^ d\) [C. R. Acad. Sci., Paris, Sér. I 301, 463-466 (1985; Zbl 0585.12013)]. Here he gives more details of that proof and shows how it gives a method for explicitly estimating C(V). In particular, he shows that if all of the conjugates of \(\alpha\) have real part at most 1/20, then \(M(\alpha)>(1.08)^ d\). This method has been recently extended by C. J. Smyth and G. Rhin to calculate lower bounds for C(V) which are, in many cases, best possible (to appear).
Note that there is an exponent \(1/(1+2t)\) missing in the inequality following (3) on p. 55.

Reviewer: D.W.Boyd

MSC:

11R04	Algebraic numbers; rings of algebraic integers
11R09	Polynomials (irreducibility, etc.)

Keywords:

algebraic integer; Lehmer conjecture; Mahler measure; transfinite diameter