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Roots of $\{0,+1,-1\}$ Polynomials.

Borwein, Peter and Pinner, Christopher (1996) Roots of $\{0,+1,-1\}$ Polynomials. [Preprint]

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      Abstract

      For a fixed $\alpha$ we discuss how closely $\alpha$ can be approximated by a root of a $\{0,+1,-1\}$ polynomial of given degree. We show that the worst rate of approximation tends to occur for roots of unity, particularly those of small degree. For roots of unity these bounds depend on the order of vanishing, $k$, of the polynomial at $\alpha$. In particular we obtain the following. Let ${\cal B}_{N}$ denote the set of roots of all $\{0,+1,-1\}$ polynomials of degree at most $N$ and ${\cal B}_{N}(\alpha,k)$ the roots of those polynomials that have a root of order at most $k$ at $\alpha$. For a Pisot number $\alpha$ in $(1,2]$ we show that \[\min_{\beta \in {\cal B}_{N}\setminus \{ \alpha \}} |\alpha -\beta| \asymp \frac{1}{\alpha^{N}},\] and for a root of unity $\alpha$ that \[\min_{\beta \in {\cal B}_{N}(\alpha,k)\setminus \{\alpha\}} |\alpha-\beta|\a symp \frac{1}{N^{(k+1) \left\lceil \frac{1}{2}\phi(d)\right\rceil +1}}.\] We study in detail the case of $\alpha=1$, where, by far, the best approximations are real. We give fairly precise bounds on the closest real root to 1. When $k=0$ or 1 we can describe the extremal polynomials explicitly.

      Item Type: Preprint
      Additional Information: pubdom FALSE
      Uncontrolled Keywords: Mahler measure, zero-one polynomials, Pisot numbers, root separation
      Subjects: 30-xx Functions of a complex variable > 30Cxx Geometric function theory
      11-xx Number theory > 11Jxx Diophantine approximation, transcendental number theory
      Faculty: UNSPECIFIED
      Depositing User: Users 1 not found.
      Date Deposited: 24 Nov 2003
      Last Modified: 21 Apr 2010 11:13
      URI: https://docserver.carma.newcastle.edu.au/id/eprint/169

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