Heights of roots of polynomials with odd coefficients

J. Garza; M. I. M. Ishak; M. J. Mossinghoff; C. G. Pinner; B. Wiles

doi:10.5802/jtnb.721

J. Garza ¹; M. I. M. Ishak ¹; M. J. Mossinghoff ²; C. G. Pinner ¹; B. Wiles ¹

¹Department of Mathematics Kansas State University Manhattan, KS 66506
²Department of Mathematics Davidson College Davidson, NC 28035-6996

Journal de théorie des nombres de Bordeaux, Volume 22 (2010) no. 2, pp. 369-381

Abstract (Original language)
Résumé

Let $α$ be a zero of a polynomial of degree $n$ with odd coefficients, with $α$ not a root of unity. We show that the height of $α$ satisfies

h (α) \geq \frac{0.4278}{n + 1} .

More generally, we obtain bounds when the coefficients are all congruent to $1$ modulo $m$ for some $m \geq 2$ .

Soit $α$ un zero d’un polynôme de degré $n$ à coefficients impairs qui n’est pas une racine de l’unité. Nous montrons que la hauteur de $α$ satisfait

h (α) \geq \frac{0.4278}{n + 1} .

Plus généralement, nous obtenons des bornes dans le cas où chaque coefficient est congru à $1$ modulo $m$ , avec $m \geq 2$ .

Article information
Cite this article

Received: 2009-05-23
Revised: 2009-12-14
Published online: 2011-01-24

MR Zbl

DOI: 10.5802/jtnb.721

Classification: 11R09, 11C08, 11R06
Keywords: Heights, Mahler measure, Lehmer’s problem

Author's affiliations:

J. Garza ¹ ; M. I. M. Ishak ¹ ; M. J. Mossinghoff ² ; C. G. Pinner ¹ ; B. Wiles ¹

¹ Department of Mathematics Kansas State University Manhattan, KS 66506
² Department of Mathematics Davidson College Davidson, NC 28035-6996

J. Garza; M. I. M. Ishak; M. J. Mossinghoff; C. G. Pinner; B. Wiles. Heights of roots of polynomials with odd coefficients. Journal de théorie des nombres de Bordeaux, Volume 22 (2010) no. 2, pp. 369-381. doi: 10.5802/jtnb.721

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References
Cited by

[1] P. Borwein, E. Dobrowolski, and M. J. Mossinghoff, Lehmer’s problem for polynomials with odd coefficients. Ann. of Math. (2) 166 (2007), no. 2, 347–366. | MR | Zbl

[2] A. Dubickas and M. J. Mossinghoff, Auxiliary polynomials for some problems regarding Mahler’s measure. Acta Arith. 119 (2005), no. 1, 65–79. | MR | Zbl

[3] M. I. M. Ishak, M. J. Mossinghoff, C. G. Pinner, and B. Wiles, Lower bounds for heights in cyclotomic extensions. J. Number Theory 130 (2010), no. 6, 1408–1424. | MR | Zbl

[4] V. A. Lebesgue, Sur l’impossibilité, en nombres entiers, de l’équation $x^{m} = y^{2} + 1$ . Nouv. Ann. Math. (1) 9 (1850), 178–181. | Numdam

[5] W. Ljunggren, Einige Bemerkungen über die Darstellung ganzer Zahlen durch binäre kubische Formen mit positiver Diskriminante. Acta Math. 75 (1943), 1–21. | MR

[6] T. Nagell, Des équations indéterminées $x^{2} + x + 1 = y^{n}$ et $x^{2} + x + 1 = 3 y^{n}$ . Norsk Matematisk Forening, Skr. Ser. I (1921), no. 2, 1–14.

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