Minorations de la maison et de la mesure de Mahler de certains entiers algébriques. (Lower bounds for the greatest absolute value of conjugates and for the Mahler’s measure of some algebraic integers). (French) Zbl 0604.12001

Langevin, Michel

C. R. Acad. Sci., Paris, Sér. I 303, 523-526 (1986).

If P is a polynomial with complex coefficients, then M(P) (Mahler’s measure of P) denotes the gometric mean on the unit circle of the modulus of P. The author shows (in paragraph 4) how to obtain explicit lower bounds on M(P) when P is monic, has integer coefficients and has all its zeros in a sector of angle strictly less than 2.
A number of other special results are proved. For example, he shows that if \(x_ 1<...<x_ d\) are all the conjugates of a totally real algebraic integer of degree at least 2 then \(x_ d-[x_ 1]\geq (3+5^{1/2})/2,\) \([x_ d]-x_ 1\geq (1+5^{1/2})/2\) and \(x_ d-x_ 1\geq 5^{1/2},\) the latter being a result of R. M. Robinson.

Reviewer: D.W.Boyd

MSC:

11R09	Polynomials (irreducibility, etc.)
12D10	Polynomials in real and complex fields: location of zeros (algebraic theorems)
11R06	PV-numbers and generalizations; other special algebraic numbers; Mahler measure

Keywords:

conjugates of algebraic integers; Mahler’s measure