A. Lemma 2.1
  [ok] (m,p)=(2,2): det[V;L] = kappa * sum_I V_I Y_I with kappa = det(LL^T)/det[Y^T|L^T] != 0
  [ok] (m,p)=(2,3): det[V;L] = kappa * sum_I V_I Y_I with kappa = det(LL^T)/det[Y^T|L^T] != 0
  [ok] (m,p)=(3,2): det[V;L] = kappa * sum_I V_I Y_I with kappa = det(LL^T)/det[Y^T|L^T] != 0
  [ok] (m,p)=(3,3): det[V;L] = kappa * sum_I V_I Y_I with kappa = det(LL^T)/det[Y^T|L^T] != 0
B. Step 4: det(zI-A-BKC) = P_{M_Sigma, y(K)}
  [ok] (m,p)=(2,2), n=4: the y(K) of 6 random gains are independent, and the linear form fitted on them reproduces det(zI-A-BKC) for 6 further gains
  [ok] (m,p)=(2,3), n=6: the y(K) of 10 random gains are independent, and the linear form fitted on them reproduces det(zI-A-BKC) for 6 further gains
  [ok] (m,p)=(3,2), n=6: the y(K) of 10 random gains are independent, and the linear form fitted on them reproduces det(zI-A-BKC) for 6 further gains
C. Step 5 for the printed matrices
  [ok] V24, eps=1/3: leading coefficient of W is V(eps) of rank m: True; W_I = T_eps(V_I): True; Wg = [D|N] with D monic: True; det(sI-A0-B0 K C0) = sum_I (Wg)_I y(K)_I: True
  [ok] V24, eps=1/7: leading coefficient of W is V(eps) of rank m: True; W_I = T_eps(V_I): True; Wg = [D|N] with D monic: True; det(sI-A0-B0 K C0) = sum_I (Wg)_I y(K)_I: True
  [ok] V33, eps=1/3: leading coefficient of W is V(eps) of rank m: True; W_I = T_eps(V_I): True; Wg = [D|N] with D monic: True; det(sI-A0-B0 K C0) = sum_I (Wg)_I y(K)_I: True
  [ok] V33, eps=1/7: leading coefficient of W is V(eps) of rank m: True; W_I = T_eps(V_I): True; Wg = [D|N] with D monic: True; det(sI-A0-B0 K C0) = sum_I (Wg)_I y(K)_I: True
  [ok] V25, eps=1/3: leading coefficient of W is V(eps) of rank m: True; W_I = T_eps(V_I): True; Wg = [D|N] with D monic: True; det(sI-A0-B0 K C0) = sum_I (Wg)_I y(K)_I: True
  [ok] V25, eps=1/7: leading coefficient of W is V(eps) of rank m: True; W_I = T_eps(V_I): True; Wg = [D|N] with D monic: True; det(sI-A0-B0 K C0) = sum_I (Wg)_I y(K)_I: True
D. Remark 3.3, the case (2,2)
  [ok] det(sI - F - GKH) = det[V'(s); K I_2] as a polynomial identity in s and the entries of K
     det(sI-F-GKH) from the printed F, G, H, by powers of s:
       s^4: +1*x(0, 0, 0, 0)
       s^3: -1*x(0, 0, 0, 1) -1*x(1, 0, 0, 0)
       s^2: -1*x(0, 1, 1, 0) +1*x(1, 0, 0, 1)
       s^1: -1*x(0, 0, 0, 0) +1*x(0, 0, 1, 0) -1*x(0, 1, 0, 0)
       s^0: +1*x(0, 0, 0, 0) +1*x(1, 0, 0, 0)
  [ok] (4.7) as printed in [BA84] differs from this expansion (signs), as Remark 3.3 says
     V' (z - 1): dim U = 2; Gram matrix of Pf on U = [['1', '1/2'], ['1/2', '1']]; det = 3/4; Pf definite on U (property (NS)): True
     matrix printed in the problem file (z + 1): dim U = 2; Gram matrix of Pf on U = [['1', '1/2'], ['1/2', '-1']]; det = -5/4; Pf definite on U (property (NS)): False
  [ok] V' has property (NS); the printed matrix does not
  [ok] equations for V': y12 = c, y23 = y14 = -c, y34 = 0, y13 + y24 = c
  [ok] equations for the printed matrix: y12 = y23 = y14 = c, y34 = 0, y24 - y13 = c
  [ok] det of the 4x4 matrix with rows (z^2,1,z,0), (z+1,z^2,1,z), (1,-t,1,0), (t+1,-1,0,1) is z^4 when t^2+t-1=0
     (the determinant as a polynomial in z, t is {(4, 0): 1, (2, 0): -1, (2, 2): 1, (2, 1): 1})
E. Example 4.5
  [ok] (a) V_R = [[1, z^2, z, 0], [-z^2, 1, 0, z]]
     (a) v = (1 + i z^2, z): dim U = 2; Gram matrix of Pf on U = [['1', '0'], ['0', '1']]; det = 1; Pf definite on U (property (NS)): True
  [ok] (a) property (NS) holds, and K_v = 0 (rank 4 of the 4 equations)
  [ok] (b) V_R as printed
  [ok] (b) H_0 is Hermitian and v H_0 v^* = 2 z^8
  [ok] (b) the 8 linear equations for K_v have rank 8 (so dim K_v = 1)
  [ok] (b) det H_0 = -20 (computed (-20+0i))
F. the degree d(m,p)
  [ok] d(2,2), d(2,4), d(2,5), d(3,3) = 2, 14, 42, 42
  [ok] d(m,p) is odd iff min = 1, or min = 2 and max = 2^k - 1, for all m, p <= 40; d(m,p) = d(p,m)
G. Wronskian of (z+1)^m, ..., (z+1)^(m+p-1)
  [ok] W = c (z+1)^(mp) with c != 0 for (m,p) = (2,2), (2,3), (3,3), (4,2), (2,5)
ALL: PASS
