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Mathematics of Computation

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The smallest Perron numbers

Author: Qiang Wu
Journal: Math. Comp. 79 (2010), 2387-2394
MSC (2010): Primary 11C08, 11R06, 11Y40
Published electronically: April 26, 2010
MathSciNet review: 2684371
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Abstract: A Perron number is a real algebraic integer $ \mathbf{\alpha} $ of degree $ d \geq 2$, whose conjugates are $ \mathbf{\alpha} _{i}$, such that $ \mathbf{\alpha} >\max _{2 \leq i \leq d} \vert \mathbf{\alpha} _{i} \vert $. In this paper we compute the smallest Perron numbers of degree $ d \leq 24$ and verify that they all satisfy the Lind-Boyd conjecture. Moreover, the smallest Perron numbers of degree 17 and 23 give the smallest house for these degrees. 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}$

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Additional Information

Qiang Wu
Affiliation: Department of Mathematics, Southwest University of China, 2 Tiansheng Road Beibei, 400715 Chongqing, China

Keywords: Algebraic integer, maximal modulus, Perron numbers, explicit auxiliary functions, integer transfinite diameter
Received by editor(s): June 9, 2009
Received by editor(s) in revised form: August 21, 2009
Published electronically: April 26, 2010
Additional Notes: This Project was supported by the Natural Science Foundation of Chongqing grant CSTC no. 2008BB0261
Article copyright: © Copyright 2010 American Mathematical Society