Hostname: page-component-7dd5485656-glrdx Total loading time: 0 Render date: 2025-10-30T22:01:33.845Z Has data issue: false hasContentIssue false
  • English
  • Français

Polynomials With {0, +1, -1} Coefficients and a Root Close to a Given Point

Published online by Cambridge University Press:  20 November 2018

Peter Borwein  and
Christopher Pinner
Peter Borwein
Affiliation:
Centre for Experimental and Constructive Mathematics, Simon Fraser University, Burnaby, BC, Canada
Christopher Pinner
Affiliation:
Centre for Experimental and Constructive Mathematics, Simon Fraser University, Burnaby, BC, Canada
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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.

Keywords

11J6830C10Mahler measurezero one polynomialsPisot numbersroot separation

Information

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 49 , Issue 5 , 01 October 1997 , pp. 887 - 915
Copyright
Copyright © Canadian Mathematical Society 1997

References

1. Barnsley, W., Fractals Everywhere. Academic Press, 1988.Google Scholar
2. Barnsley, W. and Harrington, A., A Mandelbrot set for pairs of linear maps. Physica 15D(1985), 421–432.Google Scholar
3. Beaucoup, F., Borwein, P., Boyd, D.W. and Pinner, C., Multiple roots of [-1, 1] power series. J. London Math. Soc., to appear.Google Scholar
4. Bombieri, E. and Vaaler, J.D., Polynomials with low height and prescribed vanishing. Progr. Math 70(1987), 53–73.Google Scholar
5. Borwein, P., Erdélyi, T. and Kós, G., Littlewood-type problems on [0, 1]. To appear.Google Scholar
6. Boyd, D.W., On a problem of Byrnes concerning polynomials with restricted coefficients. Math. Comp. 66(1997), 1697–1703.Google Scholar
7. Korkin, A.N. and Zolotarev, E.I., Sur un certain minimum. Nouv. Ann.Math., Sér. 2, 12(1873), 337–355.Google Scholar
8. Littlewood, J.E., On polynomials Σn ±zm and Σn eα mi zm, z = e θi. J. London Math. Soc. 41(1966), 367–376.Google Scholar
9. Mahler, K., On two extremal properties of polynomials. Illinois J. Math. 7(1963), 681–701.Google Scholar
10. Mignotte, M., Mathematics for Computer Algebra. Springer-Verlag, 1991.Google Scholar
11. Odlyzko, A. and Poonen, B., Zeros of polynomials with 0, 1 coefficients. Enseign. Math. (2) 39(1993), 317–348.Google Scholar
12. Parry, W., On the å-expansions of real numbers. Acta Math. Hungar. 11(1960), 401–416.Google Scholar
13. Rényi, A., Representations for real numbers and their ergodic properties. Acta Math. Hungar. 8(1957), 477–493.Google Scholar
14. Yamamoto, O., On some bounds for zeros of norm-bounded polynomials. J. Symbolic Comput. 18(1994), 403–42.Google Scholar