"Post-processing" hadn't occurred to me to try, since the whole point was to pre-process the root in order to economise finding the polynomial! But comparing original and post-processed polynomials for an actual train, only to discover that they have respectively 20 and 2 real positive roots, I now realise that the idea was a total dud. And this divisibility just presents another tantalising property. WFL On 8/27/15, Joerg Arndt <arndt@jjj.de> wrote:
* Fred Lunnon <fred.lunnon@gmail.com> [Aug 27. 2015 08:16]:
So I have this algebraic number x , a root of the polynomial, irreducible over the integers, F(X) = \sum_{0 <= i <= m} c_i X^{m-i} .
If it happens that some b divides each coefficient, I can divide c_i -> c_i/b , and the root remains unchanged: x -> x .
Or if c_i is divisible by b^i , I can divide c_i -> c_i/2^i , and the root is halved: x -> x/2 .
But suppose c_i divisible by b^|i-k| for some 0 < k < m : I wish to dispose of these superfluous factors as well, but what appropriate transformation should apply to x ?
I do not know of any "general" way to do this. Wouldn't post-processing the coefficients be the most simple way?
After that one may observe an educated guess (when lucky).
Best regards, jj
This behaviour is displayed by sunset polynomials of dilations of train [7, -3; 3, 2, 1] , for which b = 4 and k = m/3 . For example, the 8 novel sunsets of [r, s, p, t, q] = [84, -36; 36, 24, 12] = 12*[7, -3; 3, 2, 1] , after reduction h -> h' = (h/50)^2 [note typo in my earlier post!] , are the positive real roots, approximately 0.160458, 0.172003, 0.184713, 0.231571, 0.272437, 0.432110, 0.601110, 1.85694, of F(X) = 160000*X^12 + 614400*X^11 - 6979344*X^10 + 3276800*X^9 + 458782065*X^8 - 1720199520*X^7 + 2394836160*X^6 - 1705296384*X^5 + 697996800*X^4 - 171089920*X^3 + 24821760*X^2 - 1966080*X + 65536 --- note c_4 is odd.
Fred Lunnon
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