## 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.)

### Keywords:

polynomial; Mahler measure; inequality

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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