=begin
  # sample-geometry07.rb

  require "algebra"
  R = MPolynomial(Rational)
  m, b, a = R.vars("mba")
  K = LocalizedRing(R)
  a0, b0, m0 = K[a], K[b], K[m]
  
  M = SquareMatrix(K, 3)
  l0, l1, l2 = M[[0, 1, 0], [1, 1, -1], [1, 0, 0]]
  m4 = M[[a0, 1, -a0], [1, b0, -b0], [m0, -1, 0]]
  m40 = m4.dup; m40.set_row(0, l0)
  m41 = m4.dup; m41.set_row(1, l1)
  m42 = m4.dup; m42.set_row(2, l2)
  
  def mdet(m, i)
    i = i % 3
    m.minor(i, 2).determinant * (-1) ** i
  end
  
  def tris(m, i)
    pts = [0, 1, 2] - [i]
    (m.determinant)**2 / mdet(m, pts[0]) / mdet(m, pts[1])
  end
  
  u0 = (tris(m4, 0) - tris(m40, 0)).numerator
  u1 = (tris(m4, 1) - tris(m41, 1)).numerator
  u2 = (tris(m4, 2) - tris(m42, 2)).numerator
  
  puts [u0, u1, u2]
  puts
  
  [u0, u1, u2].each do |um|
    p um.factorize
  end
  puts
  
  x = u0 / a / (m*b+1)
  y = u1 / (m+a) / (b-1)
  z = u2 / (-b*a+1)
  
  p x
  p y
  p z
  puts
  
  gb = Groebner.basis([x, y, z])
  puts gb
  puts
  
  gb.each do |v|
    p v.factorize
  end
  #(-1)(-mb + m + ba^3 + ba^2 - ba - a^2)
  #(-1)(-ma + m + ba^2 - a^2)
  #(-1)(b + a - 1)(-ba + 1)(a)
  #(-1)(-ba + 1)(a)(a^2 + a - 1)
  
  # => a = -1/2 + \sqrt{5}/2
((<_|CONTENTS>))
=end