(L) no fifth powers in S_1 o F when dim S_2 o F = 3
  ok   F = 2*X^6 + 6*X^5*Y + 15*X^4*Y^2 + 20*X^3*Y^3 + 15*X^2*Y^4 + 6*X*Y^5 + 2*Y^6 : dim S_2 o F = 3 and S_1 o F has no fifth power
  ok   F = X^5*Y + Y^6 : dim S_2 o F = 3 and S_1 o F has no fifth power
  ok   F = X^4*Y^2 : dim S_2 o F = 3 and S_1 o F has no fifth power
  ok   F = X^3*Y^3 : dim S_2 o F = 3 and S_1 o F has no fifth power
  ok   F = X^6 + X^3*Y^3 + Y^6 : dim S_2 o F = 3 and S_1 o F has no fifth power
  ok   F = X^5*Y + X*Y^5 : dim S_2 o F = 3 and S_1 o F has no fifth power
  ok   F = -X^6 - 2*X^5*Y + X^4*Y^2 + 2*X^3*Y^3 - 3*X^2*Y^4 + X*Y^5 + Y^6 : dim S_2 o F = 3 and S_1 o F has no fifth power
  ok   F = 2*X^6 - 2*X^5*Y + 4*X^4*Y^2 + 3*X^3*Y^3 - X*Y^5 - 3*Y^6 : dim S_2 o F = 3 and S_1 o F has no fifth power
  ok   F = 4*X^6 - 4*X^4*Y^2 + 3*X^3*Y^3 + X^2*Y^4 - 3*Y^6 : dim S_2 o F = 3 and S_1 o F has no fifth power
  ok   F = 3*X^6 + 4*X^5*Y - 2*X^4*Y^2 + 2*X^3*Y^3 - X^2*Y^4 - 4*X*Y^5 - 2*Y^6 : dim S_2 o F = 3 and S_1 o F has no fifth power
  ok   F = 4*X^6 + 2*X^4*Y^2 + 2*X^3*Y^3 + X^2*Y^4 : dim S_2 o F = 3 and S_1 o F has no fifth power
  ok   F = -3*X^6 + 4*X^5*Y - 3*X^4*Y^2 + 4*X^3*Y^3 - X^2*Y^4 - X*Y^5 + 2*Y^6 : dim S_2 o F = 3 and S_1 o F has no fifth power
  ok   12 samples with dim S_2 o F = 3 tested
  ok   controls: X^6+Y^6 and X^5 Y (dim S_2 o F = 2) do have fifth powers
(F) the common-line family
  ok   (F) [U] if (1) and (2) hold, z o Phi = q o Phi + (element of D_3) + (terms of degree <= 2) identically (F, q, G, the binary parts of R5 and R4, the element of D_3 and R3 symbolic), and the third x,y-derivatives of Phi have binary cubic parts: the span of T_{<=2}, D_3 and the derivatives of Phi of order <= 2 is closed
  ok   (F) [U] z o : T_5 -> T_4 and T_4 -> T_3 are onto, with kernels of dimensions 6 and 5 (the binary forms): the R5 with (1) form a space of dimension 6 + 3 = 9 for every F with dim S_2 o F = 3, and the R4 with (2) an affine space of dimension 5 + 5 = 10
  ok   (F) [S] member 0 (G = Z^3): closed, Hilbert function [1, 3, 6, 5, 3, 2, 1]
  ok   (F) [S] member 1 (G = X*Y*Z): closed, Hilbert function [1, 3, 6, 5, 3, 2, 1]
  ok   (F) [S] member 2 (G = X^2*Z): closed, Hilbert function [1, 3, 6, 5, 3, 2, 1]
  ok   (F) [S] member 3 (G = X^3 + Y*Z^2): closed, Hilbert function [1, 3, 6, 5, 3, 2, 1]
  ok   (F) [S] member 4 (G = X*Z^2): closed, Hilbert function [1, 3, 6, 5, 3, 2, 1]
  ok   (F) [S] member 5 (G = random): closed, Hilbert function [1, 3, 6, 5, 3, 2, 1]
  ok   (F) [S] member 6 (G = random): closed, Hilbert function [1, 3, 6, 5, 3, 2, 1]
  ok   (F) [S] member 7 (G = random): closed, Hilbert function [1, 3, 6, 5, 3, 2, 1]
  ok   (F) [S] 8 accepted members of the common-line family
  ok   (F) [S] negative control: violating (1) gives a non-closed space or a different Hilbert function
  ok   (F) [S] negative control: violating (2) gives a non-closed space or a different Hilbert function
  ok   z o : T_4 -> T_3 is onto, so (2) is solvable for every (F, G, R5)
(A) an anchor of the common-line family
  ok   anchor member: closed with Hilbert function [1, 3, 6, 5, 3, 2, 1]
  ok   the three elements lie in Ann(M)
  ok   local length 36 (standard basis), m^N certificate (9, 36), < 42
    the triple: [626989/1080*x^6 + 3*x^4*y^2 - 2*x^3*y^3 - 3*x^2*y^4 - 3*x*y^5 - 3*x^5*z + x^4*y*z - 3*x^2*y^3*z + x*y^4*z - 2*y^5*z - x^4*z^2 - 2*x^3*y*z^2 - x^2*y^2*z^2 + 3*x*y^3*z^2 + 2*y^4*z^2 - x^3*z^3 + x^2*y*z^3 - x*y^2*z^3 - 2*y^3*z^3 + 3*x^2*z^4 + 3*x*y*z^4 + 3*y^2*z^4 + 2*x*z^5 - 3*y*z^5 - 3*z^6 - 4973/180*x^5 + 28739/360*x^4*y - 3*x^3*y^2 - 2*x^2*y^3 - y^5 + x^4*z - 3*x^3*y*z - 2*x^2*y^2*z + x*y^3*z - 3*y^4*z - 2*x^2*y*z^2 + 3*x*y^2*z^2 - 3*y^3*z^2 - 3*x^2*z^3 - 2*x*y*z^3 + x*z^4 - 3*y*z^4 - z^5 + 5/2*x^4 + 15*x^3*y - 65/2*x^2*y^2 + 3*x*y^3 + 3*y^4 + x^3*z - x^2*y*z - 2*x*y^2*z + y^3*z - x^2*z^2 - 2*x*y*z^2 - 2*y^2*z^2 + 3*x*z^3 + 2*y*z^3 + 2*z^4 - x^2*z + 6*x*y*z - 2*y^2*z + x*z^2 - y*z^2 - z^3, -163619/1080*x^6 - 3*x^5*y - 3*x^4*y^2 + x^2*y^4 - 2*x*y^5 + 3*y^6 + 3*x^3*y^2*z + 2*x^2*y^3*z - 2*x*y^4*z + 3*y^5*z + 3*x^4*z^2 + 2*x^3*y*z^2 + 3*x^2*y^2*z^2 + 2*y^4*z^2 - 3*x^3*z^3 - x^2*y*z^3 + 2*y^3*z^3 + 3*x*y*z^4 - y^2*z^4 - 2*x*z^5 + y*z^5 - 3*z^6 - 635/144*x^5 + 29593/720*x^4*y + x^3*y^2 + 3*x^2*y^3 + x*y^4 - 3*y^5 + 3*x^4*z - x^3*y*z + 3*x^2*y^2*z + 3*y^4*z - 3*x^3*z^2 - 2*x^2*y*z^2 - 3*x*y^2*z^2 + y^3*z^2 + x^2*z^3 + 3*y^2*z^3 + y*z^4 - z^5 - 1/2*x^4 + 69*x^3*y - 219/2*x^2*y^2 + x*y^3 - 3*y^4 - x^2*y*z - x*y^2*z - y^3*z - x^2*z^2 - 3*x*y*z^2 + y^2*z^2 - 3*x*z^3 + y*z^3 - 2*z^4 + 2*x^2*z + 18*x*y*z - 3*x*z^2 + 2*y*z^2 - 3*z^3, 20527/120*x^6 + x^5*y - 2*x^4*y^2 + 3*x^3*y^3 - 3*x^2*y^4 - 3*y^6 - 2*x^5*z - 2*x^4*y*z + x^3*y^2*z - 2*x^2*y^3*z + 3*x*y^4*z + 3*x^4*z^2 - x^2*y^2*z^2 + 2*y^4*z^2 - x^3*z^3 - x^2*y*z^3 - 3*x*y^2*z^3 - 2*y^3*z^3 - x^2*z^4 + x*y*z^4 + y^2*z^4 - 3*x*z^5 + 2*z^6 + 2633/40*x^5 - 8834/45*x^4*y + 3*x^3*y^2 - 3*x^2*y^3 + x*y^4 + 3*x^4*z + 3*x^3*y*z + 2*x^2*y^2*z - 2*x*y^3*z - 2*y^4*z + x^3*z^2 + x^2*y*z^2 - x*y^2*z^2 - y^3*z^2 + 3*x^2*z^3 - x*y*z^3 - 3*y^2*z^3 - 2*x*z^4 + 2*y*z^4 + z^5 + 5/2*x^4 - 95*x^3*y + 195/2*x^2*y^2 + 3*x^3*z + 2*x^2*y*z + 2*y^3*z - 2*x^2*z^2 + 3*x*y*z^2 - y^2*z^2 + x*z^3 + y*z^3 + 2*x^2*z - 6*x*y*z - 3*y^2*z - 2*x*z^2 + 2*y*z^2 + z^3]
ALL CHECKS PASSED
