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

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

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

Résumé
Abstract

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 $m \geq 2$ .

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 $m \geq 2$ .

MR Zbl

DOI : 10.5802/jtnb.721

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

Affiliations des auteurs :

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

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     title = {Heights of roots of polynomials with odd coefficients},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {369--381},
     publisher = {Universit\'e Bordeaux 1},
     volume = {22},
     number = {2},
     year = {2010},
     doi = {10.5802/jtnb.721},
     mrnumber = {2769068},
     zbl = {1239.11113},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.721/}
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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, Tome 22 (2010) no. 2, pp. 369-381. doi : 10.5802/jtnb.721. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.721/

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