Math 459, Senior Seminar
11/1/2011
Name: Shawn Garrity
Title: Marden’s Theorem
Source: Dan Kalman, An Elementary Proof of Marden’s Theorem, The American Mathematical
Monthly, Vol. 115, No. 4, (2008), 330-338.
Senior Project Ideas:
1. The more general form of Marden’s Theorem corresponds to the logarithmic derivative
of a product (𝑧 − 𝑧1 )𝑚1 (𝑧 − 𝑧2 )𝑚2 (𝑧 − 𝑧3 )𝑚3 where the mj’s are nonzero, and the
inscribed ellipse from the simplified version is instead a general conic section. Look into
Marden’s proof of this theorem and fill in any logical gaps (as Kalman found in his proof
of the simplified version).
2. The proof of this theorem rests on Marden’s use of the optical properties of conic
sections, which can also be used to show that the foci of an ellipse are isogonal
conjugates. Investigate these properties and find other interesting applications of them.
3. The ellipse described in the theorem is called the maximum ellipse, or sometimes the
Steiner Inellipse, named after Jakob Steiner. Explore the properties of the Steiner
Inellipse and search for other implications of these properties when dealing with
polynomials and their roots.