Heights of roots of polynomials with odd coefficients
Journal de théorie des nombres de Bordeaux, Volume 22 (2010) no. 2, pp. 369-381.

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

 $h\left(\alpha \right)\ge \frac{0.4278}{n+1}.$

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

Soit $\alpha$ 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 $\alpha$ satisfait

 $h\left(\alpha \right)\ge \frac{0.4278}{n+1}.$

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

DOI: 10.5802/jtnb.721
Classification: 11R09, 11C08, 11R06
Keywords: Heights, Mahler measure, Lehmer’s problem
Garza, J. 1; Ishak, M. I. M. 1; Mossinghoff, M. J. 2; Pinner, C. G. 1; Wiles, B. 1

1 Department of Mathematics Kansas State University Manhattan, KS 66506
2 Department of Mathematics Davidson College Davidson, NC 28035-6996
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Garza, J.; Ishak, M. I. M.; Mossinghoff, M. J.; Pinner, C. G.; Wiles, B. 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. http://archive.numdam.org/articles/10.5802/jtnb.721/

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