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

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)].

MSC:

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

Citations:

Zbl 0842.26011
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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
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