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Polynomials With {0, +1, -1} Coefficients and a Root Close to a Given Point
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- Journal:
- Canadian Journal of Mathematics / Volume 49 / Issue 5 / 01 October 1997
- Published online by Cambridge University Press:
- 20 November 2018, pp. 887-915
- Print publication:
- 01 October 1997
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- Article
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For a fixed algebraic number α we discuss how closely α 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 α.
In particular we obtain the following. Let BN denote the set of roots of all {0, +1, -1} polynomials of degree at most N and BN(α k) the roots of those polynomials that have a root of order at most k at α. For a Pisot number α in (1, 2] we show that
and for a root of unity α that
We study in detail the case of α = 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.
