Marden's theorem

In mathematics, Marden's theorem, named after Morris Marden but proved about 100 years earlier by Jörg Siebeck, gives a geometric relationship between the zeroes of a third-degree polynomial with complex coefficients and the zeroes of its derivative. See also geometrical properties of polynomial roots.

Statement

A cubic polynomial has three zeroes in the complex number plane, which in general form a triangle, and the Gauss–Lucas theorem states that the roots of its derivative lie within this triangle. Marden's theorem states their location within this triangle more precisely:

Suppose the zeroes z₁, z₂, and z₃ of a third-degree polynomial p(z) are non-collinear. There is a unique ellipse inscribed in the triangle with vertices z₁, z₂, z₃ and tangent to the sides at their midpoints: the Steiner inellipse. The foci of that ellipse are the zeroes of the derivative p'(z).

Proof

This proof comes from an exercise in Fritz Carlson's book “Geometri” (in Swedish, 1943).

Additional relations between root locations and the Steiner inellipse

By the Gauss–Lucas theorem, the root of the double derivative p"(z) must be the average of the two foci, which is the center point of the ellipse and the centroid of the triangle. In the special case that the triangle is equilateral (as happens, for instance, for the polynomial p(z) = z³ − 1) the inscribed ellipse becomes a circle, and the derivative of p has a double root at the center of the circle. Conversely, if the derivative has a double root, then the triangle must be equilateral (Kalman 2008a).

Generalizations

A more general version of the theorem, due to Linfield (1920), applies to polynomials p(z) = (z − a)ⁱ (z − b)^j (z − c)^k whose degree i + j + k may be higher than three, but that have only three roots a, b, and c. For such polynomials, the roots of the derivative may be found at the multiple roots of the given polynomial (the roots whose exponent is greater than one) and at the foci of an ellipse whose points of tangency to the triangle divide its sides in the ratios i : j, j : k, and k : i.

Another generalization (Parish (2006)) is to n-gons: some n-gons have an interior ellipse that is tangent to each side at the side's midpoint. Marden's theorem still applies: the foci of this midpoint-tangent inellipse are zeroes of the derivative of the polynomial whose zeroes are the vertices of the n-gon.

History

Jörg Siebeck discovered this theorem 81 years before Marden wrote about it. However, Dan Kalman titled his American Mathematical Monthly paper "Marden's theorem" because, as he writes, "I call this Marden’s Theorem because I first read it in M. Marden’s wonderful book".

Marden (1945, 1966) attributes what is now known as Marden's theorem to Siebeck (1864) and cites nine papers that included a version of the theorem. Dan Kalman won the 2009 Lester R. Ford Award of the Mathematical Association of America for his 2008 paper in the American Mathematical Monthly describing the theorem.

See also

• Bôcher's theorem for rational functions

References

• Kalman, Dan (2008a), "An Elementary Proof of Marden's Theorem", The American Mathematical Monthly, 115 (4): 330–338, doi:10.1080/00029890.2008.11920532, ISSN 0002-9890, S2CID 13222698
• Kalman, Dan (2008b), "The Most Marvelous Theorem in Mathematics", Journal of Online Mathematics and Its Applications
• Linfield, B. Z. (1920), "On the relation of the roots and poles of a rational function to the roots of its derivative", Bulletin of the American Mathematical Society, 27: 17–21, doi:10.1090/S0002-9904-1920-03350-1.
• Marden, Morris (1945), "A note on the zeroes of the sections of a partial fraction", Bulletin of the American Mathematical Society, 51 (12): 935–940, doi:10.1090/S0002-9904-1945-08470-5
• Marden, Morris (1966), Geometry of Polynomials, Mathematical Surveys, vol. 3, Providence, R.I.: American Mathematical Society; reprint of 1949 original publication{{citation}}: CS1 maint: postscript (link); 2005 pbk reprint with corrections
• Parish, James L. (2006), "On the derivative of a vertex polynomial" (PDF), Forum Geometricorum, 6: 285–288: Proposition 5
• Siebeck, Jörg (1864), "Über eine neue analytische Behandlungweise der Brennpunkte", Journal für die reine und angewandte Mathematik, 64: 175–182, ISSN 0075-4102 hathitrust link

1. Gauss–Lucas theorem
2. Complex number
3. Zero of a function
4. Marden
5. Zeros and poles
6. Bôcher's theorem
7. Morris Marden
8. List of misnamed theorems
9. Triangle
10. Steiner inellipse
11. Cubic equation
12. List of theorems
13. Geometrical properties of polynomial roots
14. Albert Marden
15. Tameness theorem
16. Mostow rigidity theorem
17. Dan Kalman
18. 3-manifold
19. Cohn's theorem
20. Convex hull