## On the absolute Mahler measure of polynomials having all zeros in a sector.(English)Zbl 0820.11064

Let $$P(z)\neq z$$ be a monic polynomial with integer coefficients, irreducible over the rationals, of degree $$d\geq 1$$, and having zeros $$\alpha_ 1, \dots, \alpha_ d$$. The absolute Mahler measure of $$P$$ is defined by $$\Omega (P)= \prod_{i=1}^ d \max(1, | \alpha_ i |^{1/d})$$. A result of M. Langevin [C. R. Acad. Sci., Paris, Sér. I 303, 523–526 (1986; Zbl 0604.12001)] implies that, for $$0\leq \theta< \pi$$, there is a constant $$c(\theta) >1$$ such that, if all zeros of $$P$$ lie in the sector $$\{z: | \arg z| \leq \theta\}$$, and $$P$$ is not cyclotomic, then $$\Omega (P)\geq c(\theta)$$.
The aim of this paper is to find the best possible (largest) value of $$c(\theta)$$ for as many values of $$\theta$$ as possible. The authors give explicit upper and lower bounds for $$c(\theta)$$. These bounds coincide for nine intervals $$[\theta_ i, \theta'_ i]$$ for which $$c(\theta)$$ is therefore known exactly. The function $$c(\theta)$$ is shown to be discontinuous at the left endpoints of these intervals. The angle $$\pi/2$$ lies in one of these intervals giving the interesting result that if $$P$$ is a monic irreducible polynomial with integer coefficients such that all its zeros have positive real part then either $$P(z)= z-1$$ or $$P(z)= z^ 2- z+1$$ or $$\Omega (P)\geq 1.12933793 \dots$$, where the latter is $$\Omega (z^ 6- 2z^ 5+ 4z^ 4- 5z^ 3+ 4z^ 2- 2z+ 1)$$, so the estimate is best possible. The corresponding $$\alpha$$ satisfies $$\alpha+ 1/\alpha= \theta_ 0^ 2$$, where $$\theta_ 0= 1.3247 \dots$$ is the smallest Pisot number.

### MSC:

 11R09 Polynomials (irreducibility, etc.) 12D10 Polynomials in real and complex fields: location of zeros (algebraic theorems)

Zbl 0604.12001
Full Text:

### References:

 [1] David W. Boyd, Variations on a theme of Kronecker, Canad. Math. Bull. 21 (1978), no. 2, 129 – 133. · Zbl 0392.12001 [2] David W. Boyd, Speculations concerning the range of Mahler’s measure, Canad. Math. Bull. 24 (1981), no. 4, 453 – 469. · Zbl 0474.12005 [3] David W. Boyd, Reciprocal polynomials having small measure, Math. Comp. 35 (1980), no. 152, 1361 – 1377. · Zbl 0447.12002 [4] Michel Langevin, Minorations de la maison et de la mesure de Mahler de certains entiers algébriques, C. R. Acad. Sci. Paris Sér. I Math. 303 (1986), no. 12, 523 – 526 (French, with English summary). · Zbl 0604.12001 [5] Michel Langevin, Calculs explicites de constantes de Lehmer, Groupe de travail en théorie analytique et élémentaire des nombres, 1986 – 1987, Publ. Math. Orsay, vol. 88, Univ. Paris XI, Orsay, 1988, pp. 52 – 68 (French). · Zbl 0678.12002 [6] D. H. Lehmer, Factorization of certain cyclotomic functions, Ann. of Math. (2) 34 (1933), no. 3, 461 – 479. · Zbl 0007.19904 [7] Maurice Mignotte, Sur un théorème de M. Langevin, Acta Arith. 54 (1989), no. 1, 81 – 86 (French). · Zbl 0641.12003 [8] Georges Rhin, Équirépartition modulo 1 des suites lacunaires, Séminaire de Théorie des Nombres, 1972 – 1973 (Univ. Bordeaux I, Talence), Exp. No. 19, Lab. Théorie des Nombres, Centre Nat. Recherche Sci., Talence, 1973, pp. 5 (French). · Zbl 0288.10016 [9] -, Approximants de Padé et mesures effectives d’irrationalité, Seminaire de Théorie des Nombres (May 1986), Progr. Math., vol. 71, Birkhäuser, Boston, 1987, pp. 155-164. [10] A. Schinzel, On the product of the conjugates outside the unit circle of an algebraic number, Acta Arith. 24 (1973), 385 – 399. Collection of articles dedicated to Carl Ludwig Siegel on the occasion of his seventy-fifth birthday. IV. · Zbl 0275.12004 [11] C. J. Smyth, On the measure of totally real algebraic integers, J. Austral. Math. Soc. Ser. A 30 (1980/81), no. 2, 137 – 149. C. J. Smyth, On the measure of totally real algebraic integers. II, Math. Comp. 37 (1981), no. 155, 205 – 208. · Zbl 0457.12001 [12] C. J. Smyth, The mean values of totally real algebraic integers, Math. Comp. 42 (1984), no. 166, 663 – 681. · Zbl 0536.12006
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.