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Pisot and salem numbers from polynomials of height one

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Thesis type
(Thesis) Ph.D.
Date created
We will be primarily concerned with two special kinds of real algebraic integers called Pisot and Salem numbers, both of which are characterized by the location of their conjugates in relation to the unit circle in the complex plane. While both types of numbers have been studied extensively for many years, certain important questions about Pisot numbers are generally better understood than corresponding questions about Salem numbers. In 1978 David Boyd, extending earlier work done by Jacques Dufresnoy and Charles Pisot in the 1950's, constructed an algorithm to generate all Pisot numbers in any given finite interval of the real line. Using this algorithm, we describe all Pisot numbers whose minimal polynomial is a Littlewood polynomial, one with {+1,-1}-coefficients. These are examples of polynomials that are said to have height 1, the height of a polynomial being simply the largest coefficient in absolute value. We show that every such Pisot number is a limit point, from both sides, of sequences of Salem numbers that are roots of Littlewood polynomials. We also consider analogous questions for another subset of height 1 polynomials, those with {0,1}-coefficients. Such polynomials, under a suitable normalization, have been called Newman polynomials. We describe all Pisot numbers whose minimal polynomial is derived from a Newman polynomial, and show that each Pisot number of this kind is also a limit, from both sides, of sequences of Salem numbers derived from Newman polynomials. Finally, we investigate some similarities and differences between the sets of Littlewood and Newman polynomials, especially in connection with their roots. One indicator of the location of these roots is the Mahler measure, which, for a monic polynomial, is defined as the product of the absolute values of those roots that lie outside the unit circle. Another indicator of the location of roots is the number that lie on the unit circle, and we investigate both types of polynomials with palindromic coefficient sequences in this regard.
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Thesis advisor: Borwein, Peter
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