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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A contraction of the Lucas polygon


Authors: Branko Curgus and Vania Mascioni
Journal: Proc. Amer. Math. Soc. 132 (2004), 2973-2981
MSC (2000): Primary 30C15; Secondary 26C10
Published electronically: May 20, 2004
MathSciNet review: 2063118
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Abstract: The celebrated Gauss-Lucas theorem states that all the roots of the derivative of a complex non-constant polynomial $p$ lie in the convex hull of the roots of $p$, called the Lucas polygon of $p$. We improve the Gauss-Lucas theorem by proving that all the nontrivial roots of $p'$ lie in a smaller convex polygon which is obtained by a strict contraction of the Lucas polygon of $p$.


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Additional Information

Branko Curgus
Affiliation: Department of Mathematics, Western Washington University, Bellingham, Washington 98225
Email: curgus@cc.wwu.edu

Vania Mascioni
Affiliation: Department of Mathematical Sciences, Ball State University, Muncie, Indiana 47306-0490
Email: vdm@bsu-cs.bsu.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-04-07231-4
PII: S 0002-9939(04)07231-4
Keywords: Roots of polynomials, critical points of polynomials, Gauss-Lucas theorem
Received by editor(s): October 29, 2002
Received by editor(s) in revised form: February 12, 2003
Published electronically: May 20, 2004
Communicated by: N. Tomczak-Jaegermann
Article copyright: © Copyright 2004 American Mathematical Society