I don't know the answer to Mike's question, but I do have two observations about the roots of p'(x) that may be of interest. Notation: if X are the n roots of p(x), let C(X) be the n-1 roots of p'(x). 1) Let e_j(x_1, .. x_n) be the normalized elementary symmetric polynomial, e_j = (n choose j)^(-1) sum_{1 <= k_1 < k_2 < ... < k_j <= n} x_{k_1} x_{k_2} .. x_{k_j} Then e_j(X) = e_j(C(X)) for 1 <= j <= n-1. In particular, e_1 is the geometric center of the points, so that is preserved by C. I would be very interested if anyone knew of a geometric interpretation for any of the other e_j. Proof sketch: p(x) = sum_j x^j (-1)^(n-j) S_j(X), where S_J is the unnormalized elementary symmetric polynomial. Then p'(x) = sum_j j x^(j-1) (-1)^(n-j) S_j(X) = n sum_k x^k (-1)^(n-k) S_k(C(X)). 2) C is invariant to linear transformations: C(a X + b) = a C(X) + b . Proof: (d/dx) p(a x + b) = a p'(ax + b) . One consequence of this is that for any point z and multiset of n-1 points Y, there is another multiset of n-1 points Z such that C(z union Z) = Y. Just coordinate shift so that z is the origin, then integrate p'(x) to get p(x) + k. But p(0) = 0, so k = 0. -Thomas C On Mon, Nov 15, 2010 at 11:30 AM, Mike Stay <metaweta@gmail.com> 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? -- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com
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