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Salem Numbers and Pisot Numbers via Interlacing

  Published:2011-08-03
 Printed: Apr 2012
  • James McKee,
    Department of Mathematics, Royal Holloway, University of London, Egham Hill, Egham, Surrey TW20 0EX, UK
  • Chris Smyth,
    School of Mathematics and Maxwell Institute for Mathematical Sciences, University of Edinburgh, Edinburgh EH9 3JZ, Scotland, UK
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Abstract

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: Salem numbers, Pisot numbers Salem numbers, Pisot numbers
MSC Classifications: 11R06 show english descriptions PV-numbers and generalizations; other special algebraic numbers; Mahler measure 11R06 - PV-numbers and generalizations; other special algebraic numbers; Mahler measure
 

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