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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 In mathematics, a polynomial is an expression (mathematics), expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addition, subtrac ...
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 of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
. 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 In complex analysis, a branch of mathematics, the Gauss–Lucas theorem gives a geometric relation between the roots of a polynomial ''P'' and the roots of its derivative ''P′''. The set of roots of a real or complex polynomial is a set of poin ...
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 edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, any three points, when non-collinear ...
with vertices , , and
tangent In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More ...
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''-dimen ...
s: the
Steiner inellipse In geometry, the Steiner inellipse,Weisstein, E. "Steiner Inellipse" — From MathWorld, A Wolfram Web Resource, http://mathworld.wolfram.com/SteinerInellipse.html. midpoint inellipse, or midpoint ellipse of a triangle is the unique ellipse insc ...
. The
foci Focus, or its plural form foci may refer to: Arts * Focus or Focus Festival, former name of the Adelaide Fringe arts festival in South Australia Film *''Focus'', a 1962 TV film starring James Whitmore * ''Focus'' (2001 film), a 2001 film based ...
of that ellipse are the zeroes of the derivative .


Additional relations between root locations and the Steiner inellipse

By the
Gauss–Lucas theorem In complex analysis, a branch of mathematics, the Gauss–Lucas theorem gives a geometric relation between the roots of a polynomial ''P'' and the roots of its derivative ''P′''. The set of roots of a real or complex polynomial is a set of poin ...
, 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 surface of the figure. The same definition extends to any o ...
of the triangle. In the special case that the triangle is equilateral (as happens, for instance, for the polynomial ) the inscribed ellipse degenerates to a circle, and the derivative of  has a
double root In mathematics, the multiplicity of a member of a multiset is the number of times it appears in the multiset. For example, the number of times a given polynomial has a root at a given point is the multiplicity of that root. The notion of multip ...
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 Daniel "Dan" Simon Kalman (born March 21, 1952 in Oakland, California) is an American mathematician and winner of nine awards for expository writing in mathematics. (Several article titles have links to online pdf's.) Education and career After ...
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, college, and high school teachers; graduate and undergraduate students; pure ...
for his 2008 paper in the
American Mathematical Monthly ''The American Mathematical Monthly'' is a mathematical journal founded by Benjamin Finkel in 1894. It is published ten times each year by Taylor & Francis for the Mathematical Association of America. The ''American Mathematical Monthly'' is an e ...
describing the theorem. A short and elementary proof of Marden’s theorem is explained in the solution of an exercise in Fritz Carlson’s book “Geometri” (in Swedish, 1943).


See also

* Bôcher's theorem for rational functions


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 Augus ...
, issn=0075-4102 , volume=64 , pages=175–182, url=http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=GDZPPN002152495}
hathitrust link
Theorems about triangles Theorems about polynomials Conic sections Theorems in complex geometry