# 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 pubdom FALSE Mahler measure, zero-one polynomials, Pisot numbers, root separation 30-xx Functions of a complex variable > 30Cxx Geometric function theory11-xx Number theory > 11Jxx Diophantine approximation, transcendental number theory UNSPECIFIED Users 1 not found. 24 Nov 2003 21 Apr 2010 11:13 https://docserver.carma.newcastle.edu.au/id/eprint/169