The second comparison is just a comparison of the coefficients (starting from the least significant/smallest degree) but the first flips the odd terms. So 3x < 2x < x^2 + 2x wrt the first order but 2x < x^2 + 2x < 3x for the latter. So it's not a simple reversal. Charles Greathouse Analyst/Programmer Case Western Reserve University On Sun, Jul 14, 2013 at 6:25 PM, Victor S. Miller <victorsmiller@gmail.com>wrote:
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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