A Contraction of the Lucas Polygon

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. Based on a simple proof of the classical Gauss-Lucas theorem from [5, Theorem 4.4.1] and an inequality from [2], we give an improvement of this theorem by showing that all the nontrivial roots of the derivative of a polynomial lie in a smaller convex polygon than predicted by the Gauss-Lucas theorem. In fact, our result is closely related to the following consideration of J. L. Walsh. In [12, §3.4] J. L. Walsh wrote: “A deleted neighborhood of an arbitrary zero of p(z) can be assigned which is known to contain no critical point of p(z). Since no critical point other than a multiple zero of p(z) can lie on the boundary of the Lucas polygon (assumed non-degenerate), it follows that no critical point lies in a certain strip inside the polygon and bounded by a side of the polygon and by a line parallel to that side. We proceed to make this conclusion more precise under certain conditions: [...]” However, Walsh’s results as described in [12] improve the Lucas polygon only along sides spanned by two roots of the same multiplicity and not containing any other root in the interior. Also, the formulas and calculations required in order to practically compute his improvement seem prohibitive. A significant improvement of the Gauss-Lucas theorem was achieved by D. Dimitrov in [3]. The regions that are guaranteed to contain all the nontrivial critical points of a polynomial p in [3] are given as intersections of certain unions of circles. The main difference between the results in [3] and our result is that the regions in [3] are not easy to visualize, they are not convex and there is no guarantee that these regions are strictly contained in the Gauss-Lucas polygon. In fact, a calculation of the regions studied in [3, Corollary 1] shows that in none of the cases studied in Example 3 below are these regions strictly included in the corresponding contracted Gauss-Lucas polygons constructed in this note. Therefore, the best prediction for the location of the nontrivial critical points of a polynomial is obtained as the intersection of the regions from [3] and our contracted Gauss-Lucas polygon. Received by the editors October 29, 2002 and, in revised form, February 12, 2003. 2000 Mathematics Subject Classification. Primary 30C15; Secondary 26C10.