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Salem numbers and Pisot numbers via interlacing. (English) Zbl 1333.11097

Summary: 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.

MSC:

11R06 PV-numbers and generalizations; other special algebraic numbers; Mahler measure
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