Heights of roots of polynomials with odd coefficients

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 (2010)

  • Volume: 22, Issue: 2, page 369-381
  • ISSN: 1246-7405

Abstract

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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 ) ≥ 0 . 4278 n + 1 . More generally, we obtain bounds when the coefficients are all congruent to 1 modulo m for some m ≥ 2 .

How to cite

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Garza, J., et al. "Heights of roots of polynomials with odd coefficients." Journal de Théorie des Nombres de Bordeaux 22.2 (2010): 369-381. <http://eudml.org/doc/116409>.

@article{Garza2010,
abstract = {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\[ h(\alpha )\ge \frac\{0.4278\}\{n+1\}. \]More generally, we obtain bounds when the coefficients are all congruent to $1$ modulo $m$ for some $m\ge 2$.},
affiliation = {Department of Mathematics Kansas State University Manhattan, KS 66506; Department of Mathematics Kansas State University Manhattan, KS 66506; Department of Mathematics Davidson College Davidson, NC 28035-6996; Department of Mathematics Kansas State University Manhattan, KS 66506; Department of Mathematics Kansas State University Manhattan, KS 66506},
author = {Garza, J., Ishak, M. I. M., Mossinghoff, M. J., Pinner, C. G., Wiles, B.},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {Heights; Mahler measure; Lehmer’s problem; polynomials with odd coefficients; roots; height},
language = {eng},
number = {2},
pages = {369-381},
publisher = {Université Bordeaux 1},
title = {Heights of roots of polynomials with odd coefficients},
url = {http://eudml.org/doc/116409},
volume = {22},
year = {2010},
}

TY - JOUR
AU - Garza, J.
AU - Ishak, M. I. M.
AU - Mossinghoff, M. J.
AU - Pinner, C. G.
AU - Wiles, B.
TI - Heights of roots of polynomials with odd coefficients
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2010
PB - Université Bordeaux 1
VL - 22
IS - 2
SP - 369
EP - 381
AB - 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\[ h(\alpha )\ge \frac{0.4278}{n+1}. \]More generally, we obtain bounds when the coefficients are all congruent to $1$ modulo $m$ for some $m\ge 2$.
LA - eng
KW - Heights; Mahler measure; Lehmer’s problem; polynomials with odd coefficients; roots; height
UR - http://eudml.org/doc/116409
ER -

References

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  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. Zbl1172.11034MR2373144
  2. A. Dubickas and M. J. Mossinghoff, Auxiliary polynomials for some problems regarding Mahler’s measure. Acta Arith. 119 (2005), no. 1, 65–79. Zbl1074.11018MR2163518
  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. Zbl1203.11072MR2643901
  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. 
  5. W. Ljunggren, Einige Bemerkungen über die Darstellung ganzer Zahlen durch binäre kubische Formen mit positiver Diskriminante. Acta Math. 75 (1943), 1–21. Zbl0060.09104MR17303
  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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