Inequalities on the Mahler measure of a polynomial. (Inégalités sur la mesure de Mahler d’un polynôme.) (French) Zbl 0892.11035

Flammang, V.

J. Théor. Nombres Bordx. 9, No. 1, 69-74 (1997).

The author proves the following inequalities for the Mahler measure \(M(P)\) which can be used to improve results of Bertin and Schinzel: Let \(P(x)\) be a monic polynomial with real coefficients, with all roots real and positive and with \(| P(1)| \geq 1\); then \[ M(P)^{1/d} \geq (1 + (4| P(0)| ^{1/d} + 1)^{1/2})/2. \] If instead, all roots are real and nonzero and \(| P(\pm 1)| \geq 1\) then \[ M(P)^{2/d} \geq (1 + (4| P(0)| ^{2/d} + 1)^{1/2})/2. \] The proofs use the auxiliary function \(f(x) = \log^+(x) - (c\log| x| + (1 - 2c)\log| x-1| )\). Some related inequalities established by similar techniques may be found in the author’s article [V. Flammang, Can. Math. Bull. 38, 438-444 (1995; Zbl 0842.26011)].

Reviewer: D.W.Boyd (Vancouver)

MSC:

11R09	Polynomials (irreducibility, etc.)
26C05	Real polynomials: analytic properties, etc.
12E05	Polynomials in general fields (irreducibility, etc.)

Keywords:

polynomial; Mahler measure; inequality

Citations:

Zbl 0842.26011

References:

[1]	Bertin, M.-J., Exposé au Cinquième Congrès de l’Association Canadienne de Théorie des Nombres (Ottawa, août 1996).
[2]	Schinzel, A., On the product of the conjugates outside the unit circle of an algebraic integer, Acta Arith.24 (1973), 385-399. · Zbl 0275.12004