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Salem Numbers and Pisot Numbers via Interlacing

Published online by Cambridge University Press:  20 November 2018

James McKee  and
Chris Smyth
James McKee
Affiliation:
Department of Mathematics, Royal Holloway, University of London, Egham Hill, Egham, Surrey TW20 0EX, UK email: James.McKee@rhul.ac.uk
Chris Smyth
Affiliation:
School of Mathematics and Maxwell Institute for Mathematical Sciences, University of Edinburgh, Edinburgh EH9 3JZ, Scotland, UK email: C.Smyth@ed.ac.uk
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Abstract

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We present a general construction of Salem numbers via rational functions whose zeros and poles mostly lie on the unit circle and satisfy an interlacing condition. This extends and unifies earlier work. We then consider the “obvious” limit points of the set of Salem numbers produced by our theorems and show that these are all Pisot numbers, in support of a conjecture of Boyd. We then show that all Pisot numbers arise in this way. Combining this with a theorem of Boyd, we produce all Salem numbers via an interlacing construction.

Keywords

11R06Salem numbersPisot numbers

Information

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 64 , Issue 2 , 16 April 2012 , pp. 345 - 367
Copyright
Copyright © Canadian Mathematical Society 2012

References

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