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Salem numbers of negative trace

Author: C. J. Smyth
Journal: Math. Comp. 69 (2000), 827-838
MSC (1991): Primary 11R06
Published electronically: March 10, 1999
MathSciNet review: 1648407
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Abstract: We prove that, for all $d\geq 4$, there are Salem numbers of degree $2d$ and trace $-1$, and that the number of such Salem numbers is $\gg d/\left( \log \log d\right) ^{2}$. As a consequence, it follows that the number of totally positive algebraic integers of degree $d$ and trace $2d-1$ is also $\gg d/\left( \log \log d\right) ^{2}$.

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Additional Information

C. J. Smyth
Affiliation: Department of Mathematics and Statistics, James Clerk Maxwell Building, King’s Buildings, University of Edinburgh, Mayfield Road, Edinburgh, EH9 3JZ, Scotland, UK.

Received by editor(s): April 28, 1998
Published electronically: March 10, 1999
Article copyright: © Copyright 2000 American Mathematical Society

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