for those who hava mathematica, there is a really neat interactive demo of this theorem at: http://demonstrations.wolfram.com/MardensTheorem/ bob --- Mike Stay wrote:
On Mon, Nov 15, 2010 at 7:35 AM, Henry Baker <hbaker1@pipeline.com> wrote:
I just ran across this "Most Marvelous Theorem in Mathematics":
http://www.maa.org/joma/Volume8/Kalman/index.html
It is well-known/well-taught that the roots of p'(x) lie within the convex hull of the roots of p(x).
However, in the case of a cubic, we can say a lot more: the roots of p'(x) are the _foci_ of the inscribed _ellipse_ that passes through the midpoints of the sides of the triangle formed by the roots of p(x). This is Marden's/Siebeck's theorem.
If we assume the n roots of an nth degree polynomial are algebraically independent, is the obvious generalization of this theorem true, i.e. do the roots of the derivative suffice to describe some kind of surface tangent to the sides of the n-simplex?