14 Jul
2013
14 Jul
'13
6:15 p.m.
That's what I get for writing that while sitting by the swimming pool. So given a sequence of polynomials with f(x) = 0, we have a nested sequence of partitions: first partition them by the value of the first derivative. Within one such partition, if it has more than one member, partition it by the value of the second derivative, etc. The permutations that you get must be compatible with these partitions. If the degree which actually refines a sub-partition is even, it leaves the order the same when flipping signs, if it's odd, it reverses the order. There are only three relevant real values for the coefficients: >0, = 0 and < 0. Victor On Sun, Jul 14, 2013 at 7:16 PM, Dan Asimov <dasimov@earthlink.net> wrote: > Victor, > > That would be true if there were no tangencies at 0 among the polynomials. > > But with tangencies, higher derivatives would come into play. > > For instance, it's not hard to find, for each s in S_3, 3 polynomials > which undergo the permutation s as they pass 0. > > (Or were you referring to all derivatives?) > > --Dan > > > On 2013-07-14, at 3:25 PM, Victor S. Miller 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: > >> > >> ----- > >> consider the problem of which monic 4th degree polynomials x^4 + a x^3 > + b x^2 + c x + d have the property that some line is tangent to the graph > at 2 different places. > >> . . . > >> . . . > >> ----- > >> _______________________________________________ > >> math-fun mailing list > >> math-fun@mailman.xmission.com > >> http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun > > > > _______________________________________________ > > math-fun mailing list > > math-fun@mailman.xmission.com > > http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun > > > _______________________________________________ > math-fun mailing list > math-fun@mailman.xmission.com > http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun >