[math-fun] multiply tangent lines to polyomials
a post from Erich Friedman <efriedma@stetson.edu> --rich --- greetings funsters, 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. there are some hard ways to think about this, equating slopes with average slopes, but the solution is obvious once you realize that these are the polynomials with 2 inflection points, so take 2 derivatives and look for 2 real roots. turns out the condition is 8b < 3a^2. the fact that c and d don't matter should be clear because of horizontal and vertical shifts. all that is background for my real question, one i am having trouble answering. which monic 6th degree polynomials x^6 + A x^5 + B x^4 + C x^3 + D x^2 + Ex + F have a line that is tangent to the graph at 3 different places? if such a line y = U x + V exists, then the polynomial x^6 + A x^5 + B x^4 + C x^3 + D x^2 + Ex + F - Ux - V is tangent to the x-axis at 3 points, which means it is of the form (x - r1)^2 * (x - r2)^2 * (x - r3)^2 for some distinct r1, r2, r3, but which A, B, C, and D accomplish that? i can't seem to solve the resulting equations. is there an easier way of doing this? erich friedman
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. . . . . . . -----
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
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:
----- 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
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
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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 >
Yep, you nailed it. All is forgiven. --Dan P.S. So, it turns out that this analysis leads to all but one permutation in S_4 -- and its inverse -- being realizable. Can anyone figure out which one this is? On 2013-07-14, at 5:15 PM, Victor Miller wrote:
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.
I wonder if your permutations are related to the ones described here: http://oeis.org/A027361/ On Sun, Jul 14, 2013 at 8:35 PM, Dan Asimov <dasimov@earthlink.net> wrote:
Yep, you nailed it. All is forgiven.
--Dan
P.S. So, it turns out that this analysis leads to all but one permutation in S_4 -- and its inverse -- being realizable. Can anyone figure out which one this is?
On 2013-07-14, at 5:15 PM, Victor Miller wrote:
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.
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I read of this problem in a not-too-distant Monthly, one devoted to courses given at a grad-level math summer school in Bremen, Germany. This particular article mentioned that 22 of the 24 permutations in S_4 could be realized, but I don't see 22 in that OEIS sequence. --Dan On 2013-07-14, at 6:54 PM, Victor Miller wrote:
I wonder if your permutations are related to the ones described here: http://oeis.org/A027361/
On Sun, Jul 14, 2013 at 8:35 PM, Dan Asimov <dasimov@earthlink.net> wrote:
Yep, you nailed it. All is forgiven.
--Dan
P.S. So, it turns out that this analysis leads to all but one permutation in S_4 -- and its inverse -- being realizable. Can anyone figure out which one this is?
On 2013-07-14, at 5:15 PM, Victor Miller wrote:
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.
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participants (5)
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Charles Greathouse -
Dan Asimov -
rcs＠xmission.com -
Victor Miller -
Victor S. Miller