Dan, I must be missing something. If you're talking about properties holding for sufficiently small x you're talking about derivatives, which, in the case of polynomials are the coefficients of x. So the only such permutation that occurs is reversal. Victor Sent from my iPhone On Jul 14, 2013, at 18:03, Dan Asimov <dasimov@earthlink.net> wrote:
I was just reading about this interesting question, that Erich's post reminded me of:
Call a real polynomial P(x) "tame" if P(0) = 0. I.e., the constant term = 0.
Given n (distinct) tame polynomials P_k(x), 1 <= k <= n, we can assume they're numbered such that for all negative x sufficiently near 0, we have
P_1(x) > P_2(x) > . . . > P_n(x).
Then there exists a unique permutation s in the symmetric group S_n such that for all positive x sufficiently near 0, we have
P_s(1)(x) > P_s(2)(x) > . . . > P_s(n)(x)
The question is: Are all permutations in S_n realizable by a judicious choice of the n polynomials?
--Dan
Erich Friedman wrote:
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