Geodesic
Polynomials With {0, +1, -1} Coefficients and a Root Close to a Given Point
Canadian journal of mathematics, Tome 49 (1997) no. 5, pp. 887-915

Voir la notice de l'article provenant de la source Cambridge University Press

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.
DOI : 10.4153/CJM-1997-047-3
Mots-clés : 11J68, 30C10, Mahler measure, zero one polynomials, Pisot numbers, root separation 
Borwein, Peter; Pinner, Christopher. Polynomials With {0, +1, -1} Coefficients and a Root Close to a Given Point. Canadian journal of mathematics, Tome 49 (1997) no. 5, pp. 887-915. doi: 10.4153/CJM-1997-047-3
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