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 ? 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