T1. Lemma 4.1
  [ok] (a,b)=(1,1): det[V_R;L]/Delta_S constant (0.1716): True; (-2i)^(-a) det[S;S-bar]: True; e^(i theta) S h: True
  [ok] (a,b)=(1,2): det[V_R;L]/Delta_S constant (-0.3175): True; (-2i)^(-a) det[S;S-bar]: True; e^(i theta) S h: True
  [ok] (a,b)=(1,3): det[V_R;L]/Delta_S constant (-0.1022): True; (-2i)^(-a) det[S;S-bar]: True; e^(i theta) S h: True
  [ok] (a,b)=(2,1): det[V_R;L]/Delta_S constant (0.04581): True; (-2i)^(-a) det[S;S-bar]: True; e^(i theta) S h: True
  [ok] (a,b)=(2,2): det[V_R;L]/Delta_S constant (0.05919): True; (-2i)^(-a) det[S;S-bar]: True; e^(i theta) S h: True
  [ok] (a,b)=(2,3): det[V_R;L]/Delta_S constant (-0.02176): True; (-2i)^(-a) det[S;S-bar]: True; e^(i theta) S h: True
  [ok] (a,b)=(3,2): det[V_R;L]/Delta_S constant (0.02139): True; (-2i)^(-a) det[S;S-bar]: True; e^(i theta) S h: True
T2. types j = 2 and j = 1 (a = 2)
  [ok] b=1: j=2: Delta_S = -|det S'|^2: True; j=1: Delta_S = -Im(conj(tau_1) tau_2): True
  [ok] b=2: j=2: Delta_S = -|det S'|^2: True; j=1: Delta_S = -Im(conj(tau_1) tau_2): True
  [ok] b=3: j=2: Delta_S = -|det S'|^2: True; j=1: Delta_S = -Im(conj(tau_1) tau_2): True
T3. Lemma 4.3 (Z_nu has codimension 2 nu at the points tested)
  [ok] nu=1: points with c = -0.7, 0, 0.05; residual < 1e-8; rank of the differential = 2
  [ok] nu=2: points with c = -0.7, 0, 0.05; residual < 1e-8; rank of the differential = 4
  [ok] nu=3: points with c = -0.7, 0, 0.05; residual < 1e-8; rank of the differential = 6
  [ok] nu=4: points with c = -0.7, 0, 0.05; residual < 1e-8; rank of the differential = 8
  [ok] nu=6: points with c = -0.7, 0, 0.05; residual < 1e-8; rank of the differential = 12
  [ok] nu=8: points with c = -0.7, 0, 0.05; residual < 1e-8; rank of the differential = 16
T4. Lemma 5.1 (N_{r,d})
  [ok] F=R, (r,d)=(2,1): det Pi = c z^(rd), c != 0 (relative residual 3.6e-16); rank of the differential = 2 (rd = 2)
  [ok] F=R, (r,d)=(2,2): det Pi = c z^(rd), c != 0 (relative residual 5.5e-16); rank of the differential = 4 (rd = 4)
  [ok] F=R, (r,d)=(3,2): det Pi = c z^(rd), c != 0 (relative residual 2.5e-13); rank of the differential = 6 (rd = 6)
  [ok] F=R, (r,d)=(2,4): det Pi = c z^(rd), c != 0 (relative residual 2.6e-14); rank of the differential = 8 (rd = 8)
  [ok] F=R, (r,d)=(4,2): det Pi = c z^(rd), c != 0 (relative residual 8.5e-14); rank of the differential = 8 (rd = 8)
  [ok] F=R, (r,d)=(4,4): det Pi = c z^(rd), c != 0 (relative residual 1.2e-12); rank of the differential = 16 (rd = 16)
  [ok] F=C, (r,d)=(2,2): relative residual 2.3e-15; real rank of the differential = 8 (2rd = 8)
  [ok] F=C, (r,d)=(2,4): relative residual 2.9e-14; real rank of the differential = 16 (2rd = 16)
  [ok] det = 0, rank r-1, (r,d)=(2,2), row degrees 1: residual 6.9e-16; rank of the differential of the rd+1 coefficients = 5 (rd+1 = 5)
  [ok] det = 0, rank r-1, (r,d)=(3,2), row degrees 1: residual 2.0e-15; rank of the differential of the rd+1 coefficients = 7 (rd+1 = 7)
  [ok] det = 0, rank r-1, (r,d)=(4,2), row degrees 1: residual 1.1e-14; rank of the differential of the rd+1 coefficients = 9 (rd+1 = 9)
  [ok] det = 0, rank r-1, (r,d)=(4,4), row degrees 2: residual 8.8e-14; rank of the differential of the rd+1 coefficients = 17 (rd+1 = 17)
  [ok] det = 0, rank r-1, (r,d)=(3,3), row degrees 0: residual 7.5e-16; rank of the differential of the rd+1 coefficients = 10 (rd+1 = 10)
T5. type j = 1: triples (sigma, q_1, q_2)
  [ok] d=2: residual < 1e-7 and rank of the differential = 8 at points with c = -0.5 and c = 0
  [ok] d=4: residual < 1e-7 and rank of the differential = 16 at points with c = -0.5 and c = 0
  [ok] d=6: residual < 1e-7 and rank of the differential = 24 at points with c = -0.5 and c = 0
T6. planted solutions: differential with respect to Xi
  [ok] (a,b)=(1,1), type i: relative residual 1.9e-15; 8 real parameters of Xi, 4 equations; rank of dF/dXi = 3; expected 3 = parameters - 5 (group) = equations - 1
  [ok] (a,b)=(1,2), type i: relative residual 3.1e-14; 12 real parameters of Xi, 8 equations; rank of dF/dXi = 7; expected 7 = parameters - 5 (group) = equations - 1
  [ok] (a,b)=(1,3), type i: relative residual 1.2e-13; 16 real parameters of Xi, 12 equations; rank of dF/dXi = 11; expected 11 = parameters - 5 (group) = equations - 1
  [ok] (a,b)=(2,1), type j1: relative residual 1.4e-13; 18 real parameters of Xi, 8 equations; rank of dF/dXi = 7; expected 7 = parameters - 11 (group) = equations - 1
  [ok] (a,b)=(2,2), type j0: relative residual 5.2e-13; 32 real parameters of Xi, 16 equations; rank of dF/dXi = 15; expected 15 = parameters - 17 (group) = equations - 1
  [ok] (a,b)=(2,3), type j0: relative residual 1.0e-10; 40 real parameters of Xi, 24 equations; rank of dF/dXi = 23; expected 23 = parameters - 17 (group) = equations - 1
ALL: PASS
