On the smallest value of the maximal modulus of an algebraic integer
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- by Georges Rhin and Qiang Wu;
- Math. Comp. 76 (2007), 1025-1038
- DOI: https://doi.org/10.1090/S0025-5718-06-01958-2
- Published electronically: December 29, 2006
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Abstract:
The house of an algebraic integer of degree $d$ is the largest modulus of its conjugates. For $d\leq 28$, we compute the smallest house $>1$ of degree $d$, say $\text {m}(d)$. As a consequence we improve Matveev’s theorem on the lower bound of $\text {m}(d).$ We show that, in this range, the conjecture of Schinzel-Zassenhaus is satisfied. The minimal polynomial of any algebraic integer $\boldsymbol \alpha$ whose house is equal to $\text {m}(d)$ is a factor of a bi-, tri- or quadrinomial. The computations use a family of explicit auxiliary functions. These functions depend on generalizations of the integer transfinite diameter of some compact sets in $\mathbb {C}.$ They give better bounds than the classical ones for the coefficients of the minimal polynomial of an algebraic integer $\boldsymbol \alpha$ whose house is small.References
- David W. Boyd, The maximal modulus of an algebraic integer, Math. Comp. 45 (1985), no. 171, 243–249, S17–S20. MR 790657, DOI 10.1090/S0025-5718-1985-0790657-8
- Peter Borwein, Computational excursions in analysis and number theory, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, vol. 10, Springer-Verlag, New York, 2002. MR 1912495, DOI 10.1007/978-0-387-21652-2
- Artūras Dubickas, On a conjecture of A. Schinzel and H. Zassenhaus, Acta Arith. 63 (1993), no. 1, 15–20. MR 1201616, DOI 10.4064/aa-63-1-15-20
- Valérie Flammang, Georges Rhin, and Jean-Marc Sac-Épée, Integer transfinite diameter and polynomials with small Mahler measure, Math. Comp. 75 (2006), no. 255, 1527–1540. MR 2219043, DOI 10.1090/S0025-5718-06-01791-1
- Morris Marden, Geometry of polynomials, 2nd ed., Mathematical Surveys, No. 3, American Mathematical Society, Providence, RI, 1966. MR 225972
- E. M. Matveev, On the cardinality of algebraic integers, Mat. Zametki 49 (1991), no. 4, 152–154 (Russian); English transl., Math. Notes 49 (1991), no. 3-4, 437–438. MR 1119233, DOI 10.1007/BF01158227
- C. Batut, K. Belabas, D. Bernardi, H. Cohen and M. Olivier, GP-Pari version 2.0.12, 1998.
- A. Schinzel and H. Zassenhaus, A refinement of two theorems of Kronecker, Michigan Math. J. 12 (1965), 81–85. MR 175882
- C. J. Smyth, On the product of the conjugates outside the unit circle of an algebraic integer, Bull. London Math. Soc. 3 (1971), 169–175. MR 289451, DOI 10.1112/blms/3.2.169
- Paul Voutier, An effective lower bound for the height of algebraic numbers, Acta Arith. 74 (1996), no. 1, 81–95. MR 1367580, DOI 10.4064/aa-74-1-81-95
- Qiang Wu, On the linear independence measure of logarithms of rational numbers, Math. Comp. 72 (2003), no. 242, 901–911. MR 1954974, DOI 10.1090/S0025-5718-02-01442-4
- Q. Wu, The smallest Perron numbers (in preparation).
Bibliographic Information
- Georges Rhin
- Affiliation: UMR CNRS 7122, Département UFR MIM, Université de Metz, Ile du Saulcy, 57045 Metz Cedex 01, France
- Email: rhin@math.univ-metz.fr
- Qiang Wu
- Affiliation: Department of Mathematics, Southwest University of China, 2 Tiansheng Road Beibei, 400715 Chongqing, China
- Email: qiangwu@swu.edu.cn
- Received by editor(s): December 24, 2005
- Received by editor(s) in revised form: December 28, 2005
- Published electronically: December 29, 2006
- Additional Notes: Qiang Wu was supported in part by the Natural Science Foundation of Chongqing grant CSTC no. 2005BB8024
- © Copyright 2006
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Math. Comp. 76 (2007), 1025-1038
- MSC (2000): Primary 11C08, 11R06, 11Y40
- DOI: https://doi.org/10.1090/S0025-5718-06-01958-2
- MathSciNet review: 2291848