Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-27T23:03:30.014Z Has data issue: false hasContentIssue false
  • English
  • Français

On Integer Polynomials with Multiple Roots

Part of: Polynomials and matrices

Published online by Cambridge University Press:  21 December 2009

Masaaki Amou  and
Yann Bugeaud
Masaaki Amou
Affiliation:
Department of Mathematics, Gunma University, Tenjin-cho 1–5–1, Kiryu 376–8515, Japan. E-mail: amou@math.sci.gunma-u.ac.jp
Yann Bugeaud
Affiliation:
Université Louis Pasteur, Mathématiques, 7, rue René Descartes, 67084 Strasbourg Cedex, France. E-mail: bugeaud@math.u-strasbg.fr
Get access

Abstract

A new lower bound is established for the distance between two roots of an integer polynomial, and a new upper bound for the distance between a given real number and the set of zeros of an integer polynomial. The latter result is applied to improve a metrical result in Diophantine approximation.

MSC classification

Secondary: 11C08: Polynomials
Type
Research Article
Information
Mathematika , Volume 54 , Issue 1-2 , December 2007 , pp. 83 - 92
Copyright
Copyright © University College London 2007

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Amou, M., On Sprindžuk's classification of transcendental numbers. J. Reine Angew. Math. 470 (1996), 27–50.Google Scholar
2Amou, M., Transcendence measures for almost all numbers. In Analytic Number Theory (Japanese) (Kyoto, 1995), Sūrikaisekikenkyūusho Kōkyūuroku 961 (1996), 112–116.Google Scholar
3Amou, M. and Bugeaud, Y., Sur la séparation des racines des polynômes et une question de Sprindžuk. Ramanujan J. 9 (2005), 25–32.CrossRefGoogle Scholar
4Bugeaud, Y., Approximation by Algebraic Numbers. Cambridge Tracts in Mathematics 160 (Cambridge, 2004).CrossRefGoogle Scholar
5Chudnovsky, G. V., Contributions to the Theory of Transcendental Numbers. A. M. S. Surveys and Monographs 19 (Providence, R.I., 1984).CrossRefGoogle Scholar
6Diaz, G. and Mignotte, M., Passage d'une mesure d'approximation à une mesure de transcendance. C. R. Math. Rep. Acad. Sci. Canada 13 (1991), 131–134.Google Scholar
7Güting, R., Polynomials with multiple zeros. Mathematika 14 (1967), 149–159.CrossRefGoogle Scholar
8Sprindžuk, V. G., Mahler's Problem in Metric Number Theory. American Mathematical Society (Providence, R.I., 1969).Google Scholar
9Waldschmidt, M., Diophantine Approximation on Linear Algebraic Groups. Grundlehren der mathematischen Wissenschaft 326, Springer Verlag (Berlin, Heidelberg, New York, 2000).CrossRefGoogle Scholar