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Pisot Numbers from $\{ 0, 1 \}$-Polynomials

  Published:2009-12-04
 Printed: Mar 2010
  • Keshav Mukunda
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Abstract

A \emph{Pisot number} is a real algebraic integer greater than 1, all of whose conjugates lie strictly inside the open unit disk; a \emph{Salem number} is a real algebraic integer greater than 1, all of whose conjugate roots are inside the closed unit disk, with at least one of them of modulus exactly 1. Pisot numbers have been studied extensively, and an algorithm to generate them is well known. Our main result characterises all Pisot numbers whose minimal polynomial is derived from a Newman polynomial –- one with $\{0,1\}$-coefficients –- and shows that they form a strictly increasing sequence with limit $(1+\sqrt{5}) / 2$. It has long been known that every Pisot number is a limit point, from both sides, of sequences of Salem numbers. We show that this remains true, from at least one side, for the restricted sets of Pisot and Salem numbers that are generated by Newman polynomials.
MSC Classifications: 11R06, 11R09, 11C08 show english descriptions PV-numbers and generalizations; other special algebraic numbers; Mahler measure
Polynomials (irreducibility, etc.)
Polynomials [See also 13F20]
11R06 - PV-numbers and generalizations; other special algebraic numbers; Mahler measure
11R09 - Polynomials (irreducibility, etc.)
11C08 - Polynomials [See also 13F20]
 

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