mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, 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 In mathematics, a polynomial is a Expression (mathematics), mathematical expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addit ...

with

complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...

coefficients and the zeroes of its

derivative In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is t ...

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 , , and of a third-degree polynomial are non-collinear. There is a unique ellipse inscribed in the

triangle A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called ''vertices'', are zero-dimensional points while the sides connecting them, also called ''edges'', are one-dimension ...

with vertices , , and

tangent In geometry, the tangent line (or simply tangent) to a plane curve at a given point is, intuitively, the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points o ...

to the sides at their

midpoint In geometry, the midpoint is the middle point of a line segment. It is equidistant from both endpoints, and it is the centroid both of the segment and of the endpoints. It bisects the segment. Formula The midpoint of a segment in ''n''-dim ...

s: the Steiner inellipse. The foci of that ellipse are the zeroes of the derivative .

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 must be the average of the two foci, which is the center point of the ellipse and the

centroid In mathematics and physics, the centroid, also known as geometric center or center of figure, of a plane figure or solid figure is the arithmetic mean position of all the points in the figure. The same definition extends to any object in n-d ...

of the triangle. In the special case that the triangle is equilateral (as happens, for instance, for the polynomial ) the inscribed ellipse becomes a circle, and the derivative of has a double root at the center of the circle. Conversely, if the derivative has a double root, then the triangle must be equilateral .

Generalizations

A more general version of the theorem, due to , applies to polynomials whose degree may be higher than three, but that have only three roots , , and . 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 , , and . Another generalization () 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". attributes what is now known as Marden's theorem to 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 The Mathematical Association of America (MAA) is a professional society that focuses on mathematics accessible at the undergraduate level. Members include university A university () is an educational institution, institution of tertiary edu ...

for his 2008 paper in the

American Mathematical Monthly ''The American Mathematical Monthly'' is a peer-reviewed scientific journal of mathematics. It was established by Benjamin Finkel in 1894 and is published by Taylor & Francis on behalf of the Mathematical Association of America. It is an exposi ...

describing the theorem.

References

* * * . * *
2005 pbk reprint with corrections
* * {{Citation , last1=Siebeck , first1=Jörg , title=Über eine neue analytische Behandlungweise der Brennpunkte , year=1864 , journal=

Journal für die reine und angewandte Mathematik ''Crelle's Journal'', or just ''Crelle'', is the common name for a mathematics journal, the ''Journal für die reine und angewandte Mathematik'' (in English: ''Journal for Pure and Applied Mathematics''). History The journal was founded by A ...