A232631 Coefficient table for minimal polynomials of s(2*l)^2 = (2*sin(Pi/(2*l)))^2.

-4, 1, -2, 1, -1, 1, 2, -4, 1, 1, -3, 1, 1, -4, 1, -1, 6, -5, 1, 2, -16, 20, -8, 1, -1, 9, -6, 1, 1, -12, 19, -8, 1, -1, 15, -35, 28, -9, 1, 1, -16, 20, -8, 1, 1, -21, 70, -84, 45, -11, 1, 1, -24, 86, -104, 53, -12, 1, 1, -24, 26, -9, 1, 2, -64, 336, -672, 660, -352, 104, -16, 1, 1, -36, 210, -462, 495, -286, 91, -15, 1

Offset: 1

Author

Wolfdieter Lang, Dec 18 2013

Keywords

• sign
• tabf
• easy

Comments

The length of row l of this table is delta(l) + 1 = A055034(l) + 1, l >= 1, that is: 2, 2, 2, 3, 3, 3, 4, 5, 4, 5, 6, 5, 7, 7, 5, ...

s(n):= 2*sin(Pi/n) is the length ratio side/R of a regular n-gon inscribed in a circle of radius R (in some length units). In general s(n)^2 = 4 - rho(n)^2 with rho(n):= 2*cos(Pi/n), the length ratio (smallest diagonal)/s(n) in the regular n-gon (n>=2). If n is even, say 2*l, l>=1, then s(2*l)^2 = 2 - rho(l) (because rho(2*l)^2 = rho(l) + 2). Therefore s(2*l) is an integer in the algebraic number field Q(rho(l)).

Its (monic) minimal polynomial is obtained from the conjugates of rho(l), called rho(l;j), j = 1, 2, ..., delta(l), which are the zeros of the minimal polynomial of rho(l) = rho(l;1) of degree delta(l) = A055034(l), called C(l, x) in A187360. These conjugates are therefore rho(l;j) = 2*cos(Pi*rpnodd(l,j)/l) where rpnodd(l,j) is the j-th entry of the list rpnodd(l) of the odd numbers < l which are relatively prime to l (for example, rpnodd(9) = [1,5,7], and rpnodd(9,2) = 5). From this the conjugates of s(2*l)^2 become 2 - rho(l;j), and the minimal polynomial of s(2*l)^2 is MPs2(l, x) = product( x - (2- rho(l;j)), j=1..delta(l)) for l >=1. Because the zeros of C(l, x) are integers in the algebraic number field Q(rho(l)) written in its power basis (see table 4 of the link under A187360 to the Q(2 cos(Pi/n)) paper) one finds, after expansion and reducing powers of rho(l) modulo C(l, rho(l)), directly the integer coefficients appropriate for this (monic) minimal polynomial. Only the equation C(l, rho(l)) = 0 is needed, not the trigonometric version of rho(l) and its powers.

This computation was motivated by a preprint of S. Mustonen, P. Haukkanen and J. K. Merikoski, called 'Polynomials associated with squared diagonals of regular polygons', Nov 16 2013.

Examples

			The table a(l,m) begins (n = 2*l):
  n,   l\m    0    1    2     3   4    5  6 ...
  2,   1:    -4    1
  4,   2:    -2    1
  6,   3:    -1    1
  8,   4:     2   -4    1
  10,  5:     1   -3    1
  12,  6:     1   -4    1
  14,  7:    -1    6   -5     1
  16,  8:     2  -16   20    -8   1
  18,  9:    -1    9   -6     1
  20, 10:     1  -12   19    -8   1
  22, 11:    -1   15  -35    28  -9    1
  24, 12:     1  -16   20    -8   1
  26, 13:     1  -21   70   -84  45  -11  1
  28, 14:     1  -24   86  -104  53  -12  1
  30, 15:     1  -24   26    -9   1
  ...
The minimal polynomial of s(10)^2 = (2*sin(Pi/10))^2 = 2 - rho(5) is MPs2(5, x) =  product(x - (2- rho(5;j)), j=1..2) = (x - (2 - phi))*(x - (2 - (1-phi))) with rho(5) = phi the golden section satisfying C(5, phi) = phi^2 - phi -1  = 0, hence MPs2(5, x) = 2 + phi - phi^2 - 3*x + x^2 = 1 - 3*x + x^2.
The row n=26 checks with WolframAlpha's MinimalPolynomial[(2*sin(Pi/26))^2 ,x] = 1-21 x+70 x^2-84 x^3+45 x^4-11 x^5+x^6.
		

Crossrefs

Cf. A232632 (odd n), A232633 (all n), A055034 (degree).

Programs

Mathematica

Flatten[ CoefficientList[ Table[ MinimalPolynomial[ (2*Sin[Pi/(2*l)])^2, x], {l, 1, 17}], x]] (* adapted from Jean-François Alcover, A187360 *) (* Wolfdieter Lang, Dec 23 2013 *)

Formula

a(l,m) = [x^m] MPs2(l, x), l >= 1, m = 0, 1, ...., delta(l), with the minimal polynomial MPs2(l, x) of (2*sin(Pi/(2*l)))^2, given above in a comment. The degree delta(l) = A055034(l).