Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-13T05:37:08.563Z Has data issue: false hasContentIssue false
  • English
  • Français

Bounds for the multiplicities of the roots of a complex polynomial

Part of: Polynomials and matrices Real and complex fields

Published online by Cambridge University Press:  08 April 2011

A. I. Bonciocat ,
N. C. Bonciocat  and
A. Zaharescu
A. I. Bonciocat
Affiliation:
Institute of Mathematics of the Romanian Academy, PO Box 1-764, Bucharest 014700, Romania (anca.bonciocat@imar.ro; nicolae.bonciocat@imar.ro)
N. C. Bonciocat
Affiliation:
Institute of Mathematics of the Romanian Academy, PO Box 1-764, Bucharest 014700, Romania (anca.bonciocat@imar.ro; nicolae.bonciocat@imar.ro)
A. Zaharescu
Affiliation:
Department of Mathematics, University of Illinois at Urbana-Champaign, Altgeld Hall, 1409 W. Green Street, Urbana, IL 61801, USA (zaharesc@math.uiuc.edu)
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.

We refine a result of Dubickas on the maximal multiplicity of the roots of a complex polynomial, and obtain several separability criteria for complex polynomials with large leading coefficient. We also give p-adic analogous results for polynomials with integer coefficients.

Keywords

separable polynomialmultiplicity of a rootnon-Archimedean absolute value

MSC classification

Secondary: 11C08: Polynomials 12D05: Polynomials
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 54 , Issue 3 , October 2011 , pp. 587 - 598
Copyright
Copyright © Edinburgh Mathematical Society 2011

References

1.Alkan, E., Bonciocat, A. I., Bonciocat, N. C. and Zaharescu, A., Square-free criteria for polynomials using no derivatives, Proc. Am. Math. Soc. 135(3) (2007), 677–687.CrossRefGoogle Scholar
2.Ballieu, R., Sur les limitations des racines d'une équation algébrique, Acad. R. Belg. Bull. Class. Sci. 33(5) (1947), 747–750.Google Scholar
3.Bonciocat, A. I. and Bonciocat, N. C., On the irreducibility of polynomials with leading coefficient divisible by a large prime power, Am. Math. Mon. 116(8) (2009), 743–745.CrossRefGoogle Scholar
4.Bonciocat, A. I. and Bonciocat, N. C., The irreducibility of polynomials that have one large coefficient and take a prime value, Can. Math. Bull. 52(4) (2009), 511–520.CrossRefGoogle Scholar
5.Bonciocat, A. I., Bonciocat, N. C. and Zaharescu, A., Bounds for the multiplicities of the roots for some classes of complex polynomials, Math. Inequal. Applic. 9(1) (2006), 11–22.Google Scholar
6.Dubickas, A., An inequality for the multiplicity of the roots of a polynomial, in Number theory and polynomials, London Mathematical Society Lecture Note Series, Volume 352, pp. 121–126 (Cambridge University Press, 2008).CrossRefGoogle Scholar
7.Kostrikin, A. I., Introduction to algebra (Springer, 1982).CrossRefGoogle Scholar
8.Lang, S., Algebra, Graduate Texts in Mathematics, Volume 211 (Springer, 2002).CrossRefGoogle Scholar
9.Marden, M., Geometry of polynomials, Mathematical Surveys and Monographs, Volume 3 (American Mathematical Society, Providence, RI, 1989).Google Scholar
10.Perron, O., Algebra, II: Theorie der algebraischen Gleichungen (de Gruyter, Berlin, 1951).Google Scholar
11.Waldschmidt, M., Diophantine approximation on linear algebraic groups: transcendence properties of the exponential function in several variables (Springer, 2000).CrossRefGoogle Scholar
12.Walker, R. J., Algebraic curves (Princeton University Press, 1950).Google Scholar