Marden’s Theorem

Without a doubt, my favourite theorem in all of mathematics — in the sense that it quite literally made me gape with astonishment when I first read it — is Marden’s Theorem. The result is named after Morris Marden, who publicized it in 1945 but actually attributes the result to Jörg Siebeck (1864), and provides a wonderfully simple geometric relationship between the zeroes of a third-degree polynomial with complex coefficients and the zeroes of its derivative.

Theorem. Let p: ℂ → ℂ be any polynomial of degree three with complex coefficients such that the roots z1, z2 and z3 of p form a non-trivial triangle Δ in the complex plane ℂ. Then

  1. there is a unique ellipse E (known as the Steiner inellipse) that is tangent to all three sides of Δ at their midpoints;
  2. and, furthermore, the foci of the Steiner inellipse E are precisely the zeroes of the derivative p′ of p.

To me, that’s just incredible: you start with a cubic p, associate one obvious geometric object Δ with it by the “where are the zeroes?” operation, discover that there’s another unique geometric object E associated to Δ, and the two points (foci) that equivalently define E happen to be — no, don’t just happen to be, they have to be! — the answer to “what are the zeroes of p′?”

Footnote. It turns out that the Steiner ellipse E of Δ is also the John ellipse of Δ, meaning that it is the ellipse of maximal area that is contained within Δ. Higher-dimensional convex bodies also have associated John ellipsoids of maximal volume, but I don’t know if those ellipsoids have such nice algebraic relationships with the vertices of the containing bodies.

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2 thoughts on “Marden’s Theorem”

  1. I learned a wonderful application of this result to fluid dynamics today in a fascinating talk at the Newton Institute by Morten Brøns. Place three identical point vortices in the plane, and make a triangle with them as vertices. Then the foci of the Steiner inellipse of that triangle are precisely the stagnation points of the resulting fluid flow. 😀

    See H. Araf and M. Brøns, “On stagnation points and streamline topology in vortex flows”, J. Fluid Mech. 370:1-27, 1998. http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=14363

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