roots of polynomials, critical points of polynomials, Gauss-Lucas theorem
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.
Proceedings of the American Mathematical Society
Required Publisher's Statement
First published in "Proceedings of the American Mathematical Society" in 2004, published by the American Mathematical Society.
Ćurgus, Branko, "A Contraction of the Lucas Polygon" (2004). Mathematics. 6.