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Lower bounds for Mahler measures of certain special polynomials. (Minorations pour les mesures de Mahler de certains polynômes particuliers.) (French) Zbl 1006.11007

Let \(P(x) = ax^d + \dots \pm b\) be a polynomial of degree \(d \geq 1\) with real coefficients and with \(a, b > 0\). Let \(p = |P(1)|\), \(q = |P(-1)|\) and \(r = |P(i)|\), and let \(M\) denote the Mahler measure of \(P\). The author proves in a very simple way a number of interesting inequalities, for example that if all of the roots of \(P\) are positive and if \(p > 0\) then \(M^{1/d} \geq (p^{1/d} + q^{1/d})/2\), and if all roots are real and \(p,q,r > 0\) then \(M^{2/d} \geq ((pq)^{1/d} + r^{2/d})/2\). He shows that these are stronger than the related results of V. Flammang [J. Théor. Nombres Bordx. 9, 69–74 (1997; Zbl 0892.11035)]. The main ingredient in the proof is the following simple consequence of the arithmetic-geometric mean inequality: if \(a_i, b_i > 0\) for \(i = 1,\dots,n\) then \[ (a_1\cdots a_n)^{1/n} + (b_1 \cdots b_n)^{1/n} \leq ((a_1+b_1)\cdots(a_n+b_n))^{1/n}. \]

MSC:

11C08 Polynomials in number theory
26D05 Inequalities for trigonometric functions and polynomials

Citations:

Zbl 0892.11035
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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] Flammang, V., Inégalités sur la mesure de Mahler d’un polynôme. Journal de Théorie des Nombres de Bordeaux9 (1997), 69-74. · Zbl 0892.11035
[3] Schinzel, A., On the product of the conjugates outside the unit circle of an algebraic. Acta Arith.24 (1973), 385-399. · Zbl 0275.12004
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