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
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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.