Christian Espíndola: An elementary proof of Marden's theorem

Lucas' theorem asserts that given a non-constant complex polynomial P(z), the roots of its derivative P'(z) lie inside the convex hull (in the complex plane) of the roots of P. In the particular case where P is a cubic with three non-collinear roots, vertices of a triangle, Marden's theorem provides a beautiful geometric characterization of the exact location of the roots of P'. Namely, the two roots of P' are precisely the two foci of the unique inscribed ellipse which is tangent to the sides of the triangle at their midpoints. In this talk we will present a completely elementary proof of Marden's theorem based on a relatively recent exposition by Dan Kalman, whose interest resides, on one hand, in that it intertwines standard topics in Euclidean and analytic geometry, linear algebra, complex analysis and polynomials, while on the other hand illustrates how beauty can show up in simple things.