(1) tangent cone at p0: T = N4(k_i^2 R_i)/s^2 in the variables (s, m_1, m_2)
  unit square            V=[(0, 0), (1, 0), (1, 1), (0, 1)] k=[1, -1, 1, -1]: T homogeneous of degree [6], T = s^2 * (form of degree 4 not divisible by s); deg N4(k_i^2 q_i(x)) = 4; parallel pairs [((1, 2), (3, 4)), ((1, 4), (2, 3))] (k_a+k_b = k_c+k_d = 0 there: True)  => OK
  rectangle              V=[(0, 0), (3, 0), (3, 1), (0, 1)] k=[3, -3, 3, -3]: T homogeneous of degree [6], T = s^2 * (form of degree 4 not divisible by s); deg N4(k_i^2 q_i(x)) = 4; parallel pairs [((1, 2), (3, 4)), ((1, 4), (2, 3))] (k_a+k_b = k_c+k_d = 0 there: True)  => OK
  parallelogram          V=[(0, 0), (5, 1), (7, 4), (2, 3)] k=[13, -13, 13, -13]: T homogeneous of degree [6], T = s^2 * (form of degree 4 not divisible by s); deg N4(k_i^2 q_i(x)) = 4; parallel pairs [((1, 2), (3, 4)), ((1, 4), (2, 3))] (k_a+k_b = k_c+k_d = 0 there: True)  => OK
  trapezoid A            V=[(0, 0), (7, 0), (5, 3), (1, 3)] k=[12, -12, 21, -21]: T homogeneous of degree [6], T = s^1 * (form of degree 5 not divisible by s); deg N4(k_i^2 q_i(x)) = 5; parallel pairs [((1, 2), (3, 4))] (k_a+k_b = k_c+k_d = 0 there: True)  => OK
  trapezoid B            V=[(0, 0), (4, 0), (3, 2), (1, 2)] k=[4, -4, 8, -8]: T homogeneous of degree [6], T = s^1 * (form of degree 5 not divisible by s); deg N4(k_i^2 q_i(x)) = 5; parallel pairs [((1, 2), (3, 4))] (k_a+k_b = k_c+k_d = 0 there: True)  => OK
  isosceles trapezoid    V=[(0, 0), (6, 0), (4, 3), (2, 3)] k=[6, -6, 18, -18]: T homogeneous of degree [6], T = s^1 * (form of degree 5 not divisible by s); deg N4(k_i^2 q_i(x)) = 5; parallel pairs [((1, 2), (3, 4))] (k_a+k_b = k_c+k_d = 0 there: True)  => OK
  bowtie, v1v4 || v2v3   V=[(0, 0), (6, 5), (6, 0), (0, 7)] k=[-30, -42, 42, 30]: T homogeneous of degree [6], T = s^1 * (form of degree 5 not divisible by s); deg N4(k_i^2 q_i(x)) = 5; parallel pairs [((1, 4), (2, 3))] (k_a+k_b = k_c+k_d = 0 there: True)  => OK
  bowtie, v1v3 || v2v4   V=[(0, 0), (1, 3), (4, 0), (3, 3)] k=[6, -12, -6, 12]: T homogeneous of degree [6], T = s^1 * (form of degree 5 not divisible by s); deg N4(k_i^2 q_i(x)) = 5; parallel pairs [((1, 3), (2, 4))] (k_a+k_b = k_c+k_d = 0 there: True)  => OK
  kite                   V=[(0, 0), (3, -2), (8, 0), (3, 2)] k=[20, -16, 12, -16]: T homogeneous of degree [6], T = s^0 * (form of degree 6 not divisible by s); deg N4(k_i^2 q_i(x)) = 6; parallel pairs [] (k_a+k_b = k_c+k_d = 0 there: True)  => OK
  generic4               V=[(0, 0), (7, 1), (5, 6), (-2, 3)] k=[41, -27, 23, -37]: T homogeneous of degree [6], T = s^0 * (form of degree 6 not divisible by s); deg N4(k_i^2 q_i(x)) = 6; parallel pairs [] (k_a+k_b = k_c+k_d = 0 there: True)  => OK
  non-convex             V=[(0, 0), (6, 0), (2, 1), (0, 5)] k=[-14, -10, 30, -6]: T homogeneous of degree [6], T = s^0 * (form of degree 6 not divisible by s); deg N4(k_i^2 q_i(x)) = 6; parallel pairs [] (k_a+k_b = k_c+k_d = 0 there: True)  => OK
  self-intersecting      V=[(0, 0), (5, 1), (1, 4), (6, 5)] k=[-19, 19, 19, -19]: T homogeneous of degree [6], T = s^2 * (form of degree 4 not divisible by s); deg N4(k_i^2 q_i(x)) = 4; parallel pairs [((1, 2), (3, 4)), ((1, 3), (2, 4))] (k_a+k_b = k_c+k_d = 0 there: True)  => OK
(2) three collinear vertices (outside Theorem 1.1): F_P = w_j^2 K, and ranks of E_d mod p = 2147483647
  v1,v2,v3 collinear   V=[(0, 0), (4, 1), (8, 2), (3, 7)]: deg F_P = [14]; F_P = w_4^2 K exactly: True; K: degree [12], 122 terms, vanishes at all 1000 points mod p: True; (d, monomials, rank, dim ker E_d) = [(11, 364, 364, 0), (12, 455, 454, 1), (13, 560, 556, 4), (14, 680, 670, 10)]  => OK
  v1,v2,v4 collinear   V=[(0, 0), (5, 5), (6, 0), (2, 2)]: deg F_P = [14]; F_P = w_3^2 K exactly: True; K: degree [12], 120 terms, vanishes at all 1000 points mod p: True; (d, monomials, rank, dim ker E_d) = [(11, 364, 364, 0), (12, 455, 454, 1), (13, 560, 556, 4), (14, 680, 670, 10)]  => OK
ALL OK: True
