Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-10T13:51:44.606Z Has data issue: false hasContentIssue false
  • English
  • Français

CONTINUITY OF ROOTS, REVISITED

Part of: Polynomials and matrices General theory in ordinary differential equations Geometric function theory Nonlinear algebraic or transcendental equations Real and complex fields

Published online by Cambridge University Press:  15 August 2018

DARIUSZ BUGAJEWSKI Open the ORCID record for DARIUSZ BUGAJEWSKI [Opens in a new window]  and
PIOTR MAĆKOWIAK Open the ORCID record for PIOTR MAĆKOWIAK [Opens in a new window]
DARIUSZ BUGAJEWSKI
Affiliation:
Optimization and Control Theory Department, Faculty of Mathematics and Computer Science, Adam Mickiewicz University in Poznań, ul. Umultowska 87, 61-614 Poznań, Poland email ddbb@amu.edu.pl
PIOTR MAĆKOWIAK*
Affiliation:
Department of Mathematical Economics, Poznań University of Economics and Business, Al. Niepodległości 10, 61-875 Poznań, Poland email piotr.mackowiak@ue.poznan.pl
*
✉piotr.mackowiak@ue.poznan.pl
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.

The aim of this note is to give a simple topological proof of the well-known result concerning continuity of roots of polynomials. We also consider a more general case with polynomials of a higher degree approaching a given polynomial. We then examine the continuous dependence of solutions of linear differential equations with constant coefficients.

Keywords

roots of polynomialscontinuity of roots and solutionslinear differential equations with constant coefficients

MSC classification

Primary: 30C15: Zeros of polynomials, rational functions, and other analytic functions (e.g. zeros of functions with bounded Dirichlet integral)
Secondary: 11C08: Polynomials 12D05: Polynomials 34A30: Linear equations and systems, general 65H04: Roots of polynomial equations
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 98 , Issue 3 , December 2018 , pp. 448 - 455
Copyright
© 2018 Australian Mathematical Publishing Association Inc. 

References

Bhatia, R., Matrix Analysis, Graduate Texts in Mathematics (Springer, Berlin–Heidelberg–New York, 1997).Google Scholar
Cuker, F. and Corbalan, A. G., ‘An alternate proof of the continuity of the roots of a polynomial’, Amer. Math. Monthly 96(4) (1989), 342–345.Google Scholar
Ćurgus, B. and Mascioni, V., ‘Roots and polynomials as homeomorphic spaces’, Expo. Math. 24 (2006), 81–95.Google Scholar
Harris, G. and Martin, C., ‘The roots of polynomials vary continuously as a function of the coefficients’, Proc. Amer. Math. Soc. 100(2) (1987), 390–392.Google Scholar
Harris, G. and Martin, C., ‘Large roots yield large coefficients: an addendum to “The roots of a polynomial vary continuously as a function of the coefficients”’, Proc. Amer. Math. Soc. 102(4) (1988), 993–994.Google Scholar
Rahman, Q. I. and Schmeisser, G., Analytic Theory of Polynomials: Critical Points, Zeros and Extremal Properties, London Mathematical Society Monographs (Oxford University Press, Oxford, 2002).Google Scholar