# Codimension 4 cyclic polytopes with details -- Constructing minimal resolutions for codimension 4 cyclic polytopes with details.

We give all the details of the unprojection construction for some codim 4 examples.

Δ(2,6):

 `i1 : R=QQ[x_1..x_6];` ```i2 : cc=cycRes(2,R,verbose=>2); ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ delta(2,{x_1, x_2, x_3, x_4, x_5}) + delta(0,{z, x_2, x_3, x_4}) -> delta(2,{x_1, x_2, x_3, x_4, x_5, x_6}) ------------------------------------------------------------------------------------------------------------------------ res(I): a_1 = | x_2x_4 x_3x_5 x_1x_4 x_2x_5 x_1x_3 | a_2 = {2} | 0 x_1 0 0 -x_5 | {2} | -x_1 0 x_2 0 0 | {2} | 0 -x_2 0 x_3 0 | {2} | 0 0 -x_3 0 x_4 | {2} | x_5 0 0 -x_4 0 | a_3 = {3} | x_2x_4 | {3} | x_3x_5 | {3} | x_1x_4 | {3} | x_2x_5 | {3} | x_1x_3 | ------------------------------------------------------------------------------------------------------------------------ res(J): b_1 = | z x_3 x_4 x_2 | b_2 = {1} | x_3 x_4 x_2 0 0 0 | {1} | -z 0 0 0 x_2 -x_4 | {1} | 0 -z 0 -x_2 0 x_3 | {1} | 0 0 -z x_4 -x_3 0 | b_3 = {2} | 0 x_2 -x_4 0 | {2} | -x_2 0 x_3 0 | {2} | x_4 -x_3 0 0 | {2} | z 0 0 x_3 | {2} | 0 z 0 x_4 | {2} | 0 0 z x_2 | b_4 = {3} | x_3 | {3} | x_4 | {3} | x_2 | {3} | -z | ------------------------------------------------------------------------------------------------------------------------ phi: {z, x_3, x_4, x_2} -> {x_1*x_5, 0, 0, 0} ------------------------------------------------------------------------------------------------------------------------ alpha_1 = {1} | 0 0 0 0 0 | {1} | 0 x_5 0 0 x_1 | {1} | 0 0 x_1 0 0 | {1} | x_4 0 0 x_5 0 | alpha_2 = {2} | 0 0 0 0 0 | {2} | 0 0 0 0 0 | {2} | 0 0 0 0 0 | {2} | 0 x_1 0 0 0 | {2} | 0 0 x_5 0 0 | {2} | 0 0 0 x_1 0 | alpha_3 = {3} | 0 | {3} | 0 | {3} | 0 | {3} | x_1x_5 | ------------------------------------------------------------------------------------------------------------------------ beta_1 = | -x_1x_5 0 0 0 | beta_2 = {2} | 0 0 0 0 0 0 | {2} | -x_1 0 0 0 0 0 | {2} | 0 -x_5 0 0 0 0 | {2} | 0 0 -x_1 0 0 0 | {2} | 0 0 0 0 0 0 | beta_3 = {3} | 0 0 -x_4 0 | {3} | -x_5 0 0 0 | {3} | 0 -x_1 0 0 | {3} | 0 0 -x_5 0 | {3} | -x_1 0 0 0 | ------------------------------------------------------------------------------------------------------------------------ u = -1 ------------------------------------------------------------------------------------------------------------------------ h_1 = 0 h_2 = 0 h_3 = 0 ------------------------------------------------------------------------------------------------------------------------ f_1 = | x_2x_4 x_3x_5 x_1x_4 x_2x_5 x_1x_3 x_6z-x_1x_5 x_6x_3 x_6x_4 x_6x_2 | f_2 = | 0 x_1 0 0 -x_5 0 0 0 0 0 0 x_6 0 0 0 0 | | -x_1 0 x_2 0 0 -x_1 0 0 0 0 0 0 x_6 0 0 0 | | 0 -x_2 0 x_3 0 0 -x_5 0 0 0 0 0 0 x_6 0 0 | | 0 0 -x_3 0 x_4 0 0 -x_1 0 0 0 0 0 0 x_6 0 | | x_5 0 0 -x_4 0 0 0 0 0 0 0 0 0 0 0 x_6 | | 0 0 0 0 0 -x_3 -x_4 -x_2 0 0 0 0 0 0 0 0 | | 0 0 0 0 0 z 0 0 0 -x_2 x_4 0 -x_5 0 0 -x_1 | | 0 0 0 0 0 0 z 0 x_2 0 -x_3 0 0 -x_1 0 0 | | 0 0 0 0 0 0 0 z -x_4 x_3 0 -x_4 0 0 -x_5 0 | f_3 = | 0 0 -x_4 0 -x_6 0 0 0 0 | | -x_5 0 0 0 0 -x_6 0 0 0 | | 0 -x_1 0 0 0 0 -x_6 0 0 | | 0 0 -x_5 0 0 0 0 -x_6 0 | | -x_1 0 0 0 0 0 0 0 -x_6 | | 0 -x_2 x_4 0 0 0 0 0 0 | | x_2 0 -x_3 0 0 0 0 0 0 | | -x_4 x_3 0 0 0 0 0 0 0 | | -z 0 0 -x_3 0 -x_1 0 0 0 | | 0 -z 0 -x_4 0 0 -x_5 0 0 | | 0 0 -z -x_2 0 0 0 -x_1 0 | | 0 0 0 0 0 x_1 0 0 -x_5 | | 0 0 0 0 -x_1 0 x_2 0 0 | | 0 0 0 0 0 -x_2 0 x_3 0 | | 0 0 0 0 0 0 -x_3 0 x_4 | | 0 0 0 0 x_5 0 0 -x_4 0 | f_4 = | -x_6x_3 | | -x_6x_4 | | -x_6x_2 | | x_6z-x_1x_5 | | x_2x_4 | | x_3x_5 | | x_1x_4 | | x_2x_5 | | x_1x_3 | ------------------------------------------------------------------------------------------------------------------------```

Δ(4,8):

(this of course also contains in particular the above calculation)

 `i3 : R=QQ[x_1..x_8];` ```i4 : cc=cycRes(4,R,verbose=>2); ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ delta(2,{z, x_2, x_3, x_4, x_5}) + delta(0,{z, x_2, x_3, x_4}) -> delta(2,{z, x_2, x_3, x_4, x_5, x_6}) ------------------------------------------------------------------------------------------------------------------------ res(I): a_1 = | x_2x_4 x_3x_5 zx_4 x_2x_5 zx_3 | a_2 = {2} | 0 z 0 0 -x_5 | {2} | -z 0 x_2 0 0 | {2} | 0 -x_2 0 x_3 0 | {2} | 0 0 -x_3 0 x_4 | {2} | x_5 0 0 -x_4 0 | a_3 = {3} | x_2x_4 | {3} | x_3x_5 | {3} | zx_4 | {3} | x_2x_5 | {3} | zx_3 | ------------------------------------------------------------------------------------------------------------------------ res(J): b_1 = | z x_3 x_4 x_2 | b_2 = {1} | x_3 x_4 x_2 0 0 0 | {1} | -z 0 0 0 x_2 -x_4 | {1} | 0 -z 0 -x_2 0 x_3 | {1} | 0 0 -z x_4 -x_3 0 | b_3 = {2} | 0 x_2 -x_4 0 | {2} | -x_2 0 x_3 0 | {2} | x_4 -x_3 0 0 | {2} | z 0 0 x_3 | {2} | 0 z 0 x_4 | {2} | 0 0 z x_2 | b_4 = {3} | x_3 | {3} | x_4 | {3} | x_2 | {3} | -z | ------------------------------------------------------------------------------------------------------------------------ phi: {z, x_3, x_4, x_2} -> {z*x_5, 0, 0, 0} ------------------------------------------------------------------------------------------------------------------------ alpha_1 = {1} | 0 0 0 0 0 | {1} | 0 x_5 0 0 z | {1} | 0 0 z 0 0 | {1} | x_4 0 0 x_5 0 | alpha_2 = {2} | 0 0 0 0 0 | {2} | 0 0 0 0 0 | {2} | 0 0 0 0 0 | {2} | 0 z 0 0 0 | {2} | 0 0 x_5 0 0 | {2} | 0 0 0 z 0 | alpha_3 = {3} | 0 | {3} | 0 | {3} | 0 | {3} | zx_5 | ------------------------------------------------------------------------------------------------------------------------ beta_1 = | -zx_5 0 0 0 | beta_2 = {2} | 0 0 0 0 0 0 | {2} | -z 0 0 0 0 0 | {2} | 0 -x_5 0 0 0 0 | {2} | 0 0 -z 0 0 0 | {2} | 0 0 0 0 0 0 | beta_3 = {3} | 0 0 -x_4 0 | {3} | -x_5 0 0 0 | {3} | 0 -z 0 0 | {3} | 0 0 -x_5 0 | {3} | -z 0 0 0 | ------------------------------------------------------------------------------------------------------------------------ u = -1 ------------------------------------------------------------------------------------------------------------------------ h_1 = 0 h_2 = 0 h_3 = 0 ------------------------------------------------------------------------------------------------------------------------ f_1 = | x_2x_4 x_3x_5 zx_4 x_2x_5 zx_3 x_6z-zx_5 x_6x_3 x_6x_4 x_6x_2 | f_2 = | 0 z 0 0 -x_5 0 0 0 0 0 0 x_6 0 0 0 0 | | -z 0 x_2 0 0 -z 0 0 0 0 0 0 x_6 0 0 0 | | 0 -x_2 0 x_3 0 0 -x_5 0 0 0 0 0 0 x_6 0 0 | | 0 0 -x_3 0 x_4 0 0 -z 0 0 0 0 0 0 x_6 0 | | x_5 0 0 -x_4 0 0 0 0 0 0 0 0 0 0 0 x_6 | | 0 0 0 0 0 -x_3 -x_4 -x_2 0 0 0 0 0 0 0 0 | | 0 0 0 0 0 z 0 0 0 -x_2 x_4 0 -x_5 0 0 -z | | 0 0 0 0 0 0 z 0 x_2 0 -x_3 0 0 -z 0 0 | | 0 0 0 0 0 0 0 z -x_4 x_3 0 -x_4 0 0 -x_5 0 | f_3 = | 0 0 -x_4 0 -x_6 0 0 0 0 | | -x_5 0 0 0 0 -x_6 0 0 0 | | 0 -z 0 0 0 0 -x_6 0 0 | | 0 0 -x_5 0 0 0 0 -x_6 0 | | -z 0 0 0 0 0 0 0 -x_6 | | 0 -x_2 x_4 0 0 0 0 0 0 | | x_2 0 -x_3 0 0 0 0 0 0 | | -x_4 x_3 0 0 0 0 0 0 0 | | -z 0 0 -x_3 0 -z 0 0 0 | | 0 -z 0 -x_4 0 0 -x_5 0 0 | | 0 0 -z -x_2 0 0 0 -z 0 | | 0 0 0 0 0 z 0 0 -x_5 | | 0 0 0 0 -z 0 x_2 0 0 | | 0 0 0 0 0 -x_2 0 x_3 0 | | 0 0 0 0 0 0 -x_3 0 x_4 | | 0 0 0 0 x_5 0 0 -x_4 0 | f_4 = | -x_6x_3 | | -x_6x_4 | | -x_6x_2 | | x_6z-zx_5 | | x_2x_4 | | x_3x_5 | | zx_4 | | x_2x_5 | | zx_3 | ------------------------------------------------------------------------------------------------------------------------ ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ delta(4,{x_1, x_2, x_3, x_4, x_5, x_6, x_7}) + delta(2,{z, x_2, x_3, x_4, x_5, x_6}) -> delta(4,{x_1, x_2, x_3, x_4, x_5, x_6, x_7, x_8}) ------------------------------------------------------------------------------------------------------------------------ res(I): a_1 = | x_2x_4x_6 x_3x_5x_7 x_1x_4x_6 x_2x_5x_7 x_1x_3x_6 x_2x_4x_7 x_1x_3x_5 | a_2 = {3} | 0 x_1 0 0 0 0 -x_7 | {3} | -x_1 0 x_2 0 0 0 0 | {3} | 0 -x_2 0 x_3 0 0 0 | {3} | 0 0 -x_3 0 x_4 0 0 | {3} | 0 0 0 -x_4 0 x_5 0 | {3} | 0 0 0 0 -x_5 0 x_6 | {3} | x_7 0 0 0 0 -x_6 0 | a_3 = {4} | x_2x_4x_6 | {4} | x_3x_5x_7 | {4} | x_1x_4x_6 | {4} | x_2x_5x_7 | {4} | x_1x_3x_6 | {4} | x_2x_4x_7 | {4} | x_1x_3x_5 | ------------------------------------------------------------------------------------------------------------------------ res(J): b_1 = | x_2x_4 x_3x_5 zx_4 x_2x_5 zx_3 -zx_5 x_3x_6 x_4x_6 x_2x_6 | b_2 = {2} | 0 z 0 0 -x_5 0 0 0 0 0 0 x_6 0 0 0 0 | {2} | -z 0 x_2 0 0 -z 0 0 0 0 0 0 x_6 0 0 0 | {2} | 0 -x_2 0 x_3 0 0 -x_5 0 0 0 0 0 0 x_6 0 0 | {2} | 0 0 -x_3 0 x_4 0 0 -z 0 0 0 0 0 0 x_6 0 | {2} | x_5 0 0 -x_4 0 0 0 0 0 0 0 0 0 0 0 x_6 | {2} | 0 0 0 0 0 -x_3 -x_4 -x_2 0 0 0 0 0 0 0 0 | {2} | 0 0 0 0 0 0 0 0 0 -x_2 x_4 0 -x_5 0 0 -z | {2} | 0 0 0 0 0 0 0 0 x_2 0 -x_3 0 0 -z 0 0 | {2} | 0 0 0 0 0 0 0 0 -x_4 x_3 0 -x_4 0 0 -x_5 0 | b_3 = {3} | 0 0 -x_4 0 -x_6 0 0 0 0 | {3} | -x_5 0 0 0 0 -x_6 0 0 0 | {3} | 0 -z 0 0 0 0 -x_6 0 0 | {3} | 0 0 -x_5 0 0 0 0 -x_6 0 | {3} | -z 0 0 0 0 0 0 0 -x_6 | {3} | 0 -x_2 x_4 0 0 0 0 0 0 | {3} | x_2 0 -x_3 0 0 0 0 0 0 | {3} | -x_4 x_3 0 0 0 0 0 0 0 | {3} | 0 0 0 -x_3 0 -z 0 0 0 | {3} | 0 0 0 -x_4 0 0 -x_5 0 0 | {3} | 0 0 0 -x_2 0 0 0 -z 0 | {3} | 0 0 0 0 0 z 0 0 -x_5 | {3} | 0 0 0 0 -z 0 x_2 0 0 | {3} | 0 0 0 0 0 -x_2 0 x_3 0 | {3} | 0 0 0 0 0 0 -x_3 0 x_4 | {3} | 0 0 0 0 x_5 0 0 -x_4 0 | b_4 = {4} | -x_3x_6 | {4} | -x_4x_6 | {4} | -x_2x_6 | {4} | -zx_5 | {4} | x_2x_4 | {4} | x_3x_5 | {4} | zx_4 | {4} | x_2x_5 | {4} | zx_3 | ------------------------------------------------------------------------------------------------------------------------ phi: {x_2*x_4, x_3*x_5, z*x_4, x_2*x_5, z*x_3, -z*x_5, x_3*x_6, x_4*x_6, x_2*x_6} -> {0, 0, x_1*x_4*x_7, 0, x_1*x_3*x_7, -x_1*x_5*x_7, 0, 0, 0} ------------------------------------------------------------------------------------------------------------------------ alpha_1 = {2} | x_6 0 0 0 0 x_7 0 | {2} | 0 x_7 0 0 0 0 x_1 | {2} | 0 0 0 0 0 0 0 | {2} | 0 0 0 x_7 0 0 0 | {2} | 0 0 0 0 0 0 0 | {2} | 0 0 0 0 0 0 0 | {2} | 0 0 0 0 x_1 0 0 | {2} | 0 0 x_1 0 0 0 0 | {2} | 0 0 0 0 0 0 0 | alpha_2 = {3} | 0 0 0 0 0 0 0 | {3} | 0 0 0 0 0 0 0 | {3} | 0 0 x_7 0 0 0 0 | {3} | 0 0 0 0 0 0 0 | {3} | 0 0 0 0 x_7 0 0 | {3} | 0 0 0 0 0 0 0 | {3} | 0 0 0 0 0 0 0 | {3} | 0 0 0 0 0 0 0 | {3} | 0 -x_1 0 0 0 0 0 | {3} | 0 0 0 0 0 0 0 | {3} | 0 0 0 -x_1 0 0 0 | {3} | 0 x_1 0 0 0 0 0 | {3} | 0 0 0 0 0 -x_1 0 | {3} | 0 0 0 0 0 0 0 | {3} | 0 0 0 0 0 0 0 | {3} | 0 0 0 0 0 0 0 | alpha_3 = {4} | 0 | {4} | 0 | {4} | 0 | {4} | x_1x_5x_7 | {4} | 0 | {4} | 0 | {4} | -x_1x_4x_7 | {4} | 0 | {4} | -x_1x_3x_7 | ------------------------------------------------------------------------------------------------------------------------ beta_1 = | 0 0 -x_1x_4x_7 0 -x_1x_3x_7 x_1x_5x_7 0 0 0 | beta_2 = {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | {3} | -x_1 0 0 0 0 -x_1 0 0 0 0 0 0 0 0 0 0 | {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 -x_7 0 0 | {3} | 0 0 0 0 0 0 0 -x_1 0 0 0 0 0 0 0 0 | {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -x_7 | {3} | 0 x_1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | beta_3 = {4} | 0 0 0 0 -x_6 0 0 0 0 | {4} | 0 0 0 0 0 -x_7 0 0 0 | {4} | 0 x_1 0 0 0 0 0 0 0 | {4} | 0 0 0 0 0 0 0 -x_7 0 | {4} | x_1 0 0 0 0 0 0 0 0 | {4} | 0 0 0 0 -x_7 0 0 0 0 | {4} | 0 0 0 0 0 -x_1 0 0 0 | ------------------------------------------------------------------------------------------------------------------------ u = -1 ------------------------------------------------------------------------------------------------------------------------ h_1 = 0 h_2 = 0 h_3 = 0 ------------------------------------------------------------------------------------------------------------------------ f_1 = | x_2x_4x_6 x_3x_5x_7 x_1x_4x_6 x_2x_5x_7 x_1x_3x_6 x_2x_4x_7 x_1x_3x_5 x_8x_2x_4 x_8x_3x_5 x_8zx_4-x_1x_4x_7 x_8x_2x_5 x_8zx_3-x_1x_3x_7 -x_8zx_5+x_1x_5x_7 x_8x_3x_6 x_8x_4x_6 x_8x_2x_6 | f_2 = | 0 x_1 0 0 0 0 -x_7 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x_8 0 0 0 0 0 0 | | -x_1 0 x_2 0 0 0 0 -x_1 0 0 0 0 -x_1 0 0 0 0 0 0 0 0 0 0 0 x_8 0 0 0 0 0 | | 0 -x_2 0 x_3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -x_7 0 0 0 0 x_8 0 0 0 0 | | 0 0 -x_3 0 x_4 0 0 0 0 0 0 0 0 0 -x_1 0 0 0 0 0 0 0 0 0 0 0 x_8 0 0 0 | | 0 0 0 -x_4 0 x_5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -x_7 0 0 0 0 x_8 0 0 | | 0 0 0 0 -x_5 0 x_6 0 x_1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x_8 0 | | x_7 0 0 0 0 -x_6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x_8 | | 0 0 0 0 0 0 0 0 -z 0 0 x_5 0 0 0 0 0 0 -x_6 0 0 0 0 -x_6 0 0 0 0 -x_7 0 | | 0 0 0 0 0 0 0 z 0 -x_2 0 0 z 0 0 0 0 0 0 -x_6 0 0 0 0 -x_7 0 0 0 0 -x_1 | | 0 0 0 0 0 0 0 0 x_2 0 -x_3 0 0 x_5 0 0 0 0 0 0 -x_6 0 0 0 0 0 0 0 0 0 | | 0 0 0 0 0 0 0 0 0 x_3 0 -x_4 0 0 z 0 0 0 0 0 0 -x_6 0 0 0 0 -x_7 0 0 0 | | 0 0 0 0 0 0 0 -x_5 0 0 x_4 0 0 0 0 0 0 0 0 0 0 0 -x_6 0 0 0 0 0 0 0 | | 0 0 0 0 0 0 0 0 0 0 0 0 x_3 x_4 x_2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x_2 -x_4 0 x_5 0 0 z 0 0 0 0 -x_1 0 0 | | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -x_2 0 x_3 0 0 z 0 0 0 0 -x_1 0 0 0 0 | | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x_4 -x_3 0 x_4 0 0 x_5 0 0 0 0 0 0 0 0 | f_3 = | 0 0 0 0 -x_6 0 0 0 0 -x_8 0 0 0 0 0 0 | | 0 0 0 0 0 -x_7 0 0 0 0 -x_8 0 0 0 0 0 | | 0 x_1 0 0 0 0 0 0 0 0 0 -x_8 0 0 0 0 | | 0 0 0 0 0 0 0 -x_7 0 0 0 0 -x_8 0 0 0 | | x_1 0 0 0 0 0 0 0 0 0 0 0 0 -x_8 0 0 | | 0 0 0 0 -x_7 0 0 0 0 0 0 0 0 0 -x_8 0 | | 0 0 0 0 0 -x_1 0 0 0 0 0 0 0 0 0 -x_8 | | 0 0 x_4 0 x_6 0 0 0 0 0 0 0 0 0 0 0 | | x_5 0 0 0 0 x_6 0 0 0 0 0 0 0 0 0 0 | | 0 z 0 0 0 0 x_6 0 0 0 0 -x_7 0 0 0 0 | | 0 0 x_5 0 0 0 0 x_6 0 0 0 0 0 0 0 0 | | z 0 0 0 0 0 0 0 x_6 0 0 0 0 -x_7 0 0 | | 0 x_2 -x_4 0 0 0 0 0 0 0 0 0 0 0 0 0 | | -x_2 0 x_3 0 0 0 0 0 0 0 0 0 0 0 0 0 | | x_4 -x_3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | | 0 0 0 x_3 0 z 0 0 0 0 x_1 0 0 0 0 0 | | 0 0 0 x_4 0 0 x_5 0 0 0 0 0 0 0 0 0 | | 0 0 0 x_2 0 0 0 z 0 0 0 0 x_1 0 0 0 | | 0 0 0 0 0 -z 0 0 x_5 0 -x_1 0 0 0 0 0 | | 0 0 0 0 z 0 -x_2 0 0 0 0 0 0 0 x_1 0 | | 0 0 0 0 0 x_2 0 -x_3 0 0 0 0 0 0 0 0 | | 0 0 0 0 0 0 x_3 0 -x_4 0 0 0 0 0 0 0 | | 0 0 0 0 -x_5 0 0 x_4 0 0 0 0 0 0 0 0 | | 0 0 0 0 0 0 0 0 0 0 x_1 0 0 0 0 -x_7 | | 0 0 0 0 0 0 0 0 0 -x_1 0 x_2 0 0 0 0 | | 0 0 0 0 0 0 0 0 0 0 -x_2 0 x_3 0 0 0 | | 0 0 0 0 0 0 0 0 0 0 0 -x_3 0 x_4 0 0 | | 0 0 0 0 0 0 0 0 0 0 0 0 -x_4 0 x_5 0 | | 0 0 0 0 0 0 0 0 0 0 0 0 0 -x_5 0 x_6 | | 0 0 0 0 0 0 0 0 0 x_7 0 0 0 0 -x_6 0 | f_4 = | x_8x_3x_6 | | x_8x_4x_6 | | x_8x_2x_6 | | x_8zx_5-x_1x_5x_7 | | -x_8x_2x_4 | | -x_8x_3x_5 | | -x_8zx_4+x_1x_4x_7 | | -x_8x_2x_5 | | -x_8zx_3+x_1x_3x_7 | | x_2x_4x_6 | | x_3x_5x_7 | | x_1x_4x_6 | | x_2x_5x_7 | | x_1x_3x_6 | | x_2x_4x_7 | | x_1x_3x_5 | ------------------------------------------------------------------------------------------------------------------------```

Δ(6,10):

(this of course also contains in particular the above calculation)

 `i5 : R=QQ[x_1..x_10];` ```i6 : cc=cycRes(6,R,verbose=>2); ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ delta(2,{z, x_2, x_3, x_4, x_5}) + delta(0,{z, x_2, x_3, x_4}) -> delta(2,{z, x_2, x_3, x_4, x_5, x_6}) ------------------------------------------------------------------------------------------------------------------------ res(I): a_1 = | x_2x_4 x_3x_5 zx_4 x_2x_5 zx_3 | a_2 = {2} | 0 z 0 0 -x_5 | {2} | -z 0 x_2 0 0 | {2} | 0 -x_2 0 x_3 0 | {2} | 0 0 -x_3 0 x_4 | {2} | x_5 0 0 -x_4 0 | a_3 = {3} | x_2x_4 | {3} | x_3x_5 | {3} | zx_4 | {3} | x_2x_5 | {3} | zx_3 | ------------------------------------------------------------------------------------------------------------------------ res(J): b_1 = | z x_3 x_4 x_2 | b_2 = {1} | x_3 x_4 x_2 0 0 0 | {1} | -z 0 0 0 x_2 -x_4 | {1} | 0 -z 0 -x_2 0 x_3 | {1} | 0 0 -z x_4 -x_3 0 | b_3 = {2} | 0 x_2 -x_4 0 | {2} | -x_2 0 x_3 0 | {2} | x_4 -x_3 0 0 | {2} | z 0 0 x_3 | {2} | 0 z 0 x_4 | {2} | 0 0 z x_2 | b_4 = {3} | x_3 | {3} | x_4 | {3} | x_2 | {3} | -z | ------------------------------------------------------------------------------------------------------------------------ phi: {z, x_3, x_4, x_2} -> {z*x_5, 0, 0, 0} ------------------------------------------------------------------------------------------------------------------------ alpha_1 = {1} | 0 0 0 0 0 | {1} | 0 x_5 0 0 z | {1} | 0 0 z 0 0 | {1} | x_4 0 0 x_5 0 | alpha_2 = {2} | 0 0 0 0 0 | {2} | 0 0 0 0 0 | {2} | 0 0 0 0 0 | {2} | 0 z 0 0 0 | {2} | 0 0 x_5 0 0 | {2} | 0 0 0 z 0 | alpha_3 = {3} | 0 | {3} | 0 | {3} | 0 | {3} | zx_5 | ------------------------------------------------------------------------------------------------------------------------ beta_1 = | -zx_5 0 0 0 | beta_2 = {2} | 0 0 0 0 0 0 | {2} | -z 0 0 0 0 0 | {2} | 0 -x_5 0 0 0 0 | {2} | 0 0 -z 0 0 0 | {2} | 0 0 0 0 0 0 | beta_3 = {3} | 0 0 -x_4 0 | {3} | -x_5 0 0 0 | {3} | 0 -z 0 0 | {3} | 0 0 -x_5 0 | {3} | -z 0 0 0 | ------------------------------------------------------------------------------------------------------------------------ u = -1 ------------------------------------------------------------------------------------------------------------------------ h_1 = 0 h_2 = 0 h_3 = 0 ------------------------------------------------------------------------------------------------------------------------ f_1 = | x_2x_4 x_3x_5 zx_4 x_2x_5 zx_3 x_6z-zx_5 x_6x_3 x_6x_4 x_6x_2 | f_2 = | 0 z 0 0 -x_5 0 0 0 0 0 0 x_6 0 0 0 0 | | -z 0 x_2 0 0 -z 0 0 0 0 0 0 x_6 0 0 0 | | 0 -x_2 0 x_3 0 0 -x_5 0 0 0 0 0 0 x_6 0 0 | | 0 0 -x_3 0 x_4 0 0 -z 0 0 0 0 0 0 x_6 0 | | x_5 0 0 -x_4 0 0 0 0 0 0 0 0 0 0 0 x_6 | | 0 0 0 0 0 -x_3 -x_4 -x_2 0 0 0 0 0 0 0 0 | | 0 0 0 0 0 z 0 0 0 -x_2 x_4 0 -x_5 0 0 -z | | 0 0 0 0 0 0 z 0 x_2 0 -x_3 0 0 -z 0 0 | | 0 0 0 0 0 0 0 z -x_4 x_3 0 -x_4 0 0 -x_5 0 | f_3 = | 0 0 -x_4 0 -x_6 0 0 0 0 | | -x_5 0 0 0 0 -x_6 0 0 0 | | 0 -z 0 0 0 0 -x_6 0 0 | | 0 0 -x_5 0 0 0 0 -x_6 0 | | -z 0 0 0 0 0 0 0 -x_6 | | 0 -x_2 x_4 0 0 0 0 0 0 | | x_2 0 -x_3 0 0 0 0 0 0 | | -x_4 x_3 0 0 0 0 0 0 0 | | -z 0 0 -x_3 0 -z 0 0 0 | | 0 -z 0 -x_4 0 0 -x_5 0 0 | | 0 0 -z -x_2 0 0 0 -z 0 | | 0 0 0 0 0 z 0 0 -x_5 | | 0 0 0 0 -z 0 x_2 0 0 | | 0 0 0 0 0 -x_2 0 x_3 0 | | 0 0 0 0 0 0 -x_3 0 x_4 | | 0 0 0 0 x_5 0 0 -x_4 0 | f_4 = | -x_6x_3 | | -x_6x_4 | | -x_6x_2 | | x_6z-zx_5 | | x_2x_4 | | x_3x_5 | | zx_4 | | x_2x_5 | | zx_3 | ------------------------------------------------------------------------------------------------------------------------ ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ delta(4,{z, x_2, x_3, x_4, x_5, x_6, x_7}) + delta(2,{z, x_2, x_3, x_4, x_5, x_6}) -> delta(4,{z, x_2, x_3, x_4, x_5, x_6, x_7, x_8}) ------------------------------------------------------------------------------------------------------------------------ res(I): a_1 = | x_2x_4x_6 x_3x_5x_7 zx_4x_6 x_2x_5x_7 zx_3x_6 x_2x_4x_7 zx_3x_5 | a_2 = {3} | 0 z 0 0 0 0 -x_7 | {3} | -z 0 x_2 0 0 0 0 | {3} | 0 -x_2 0 x_3 0 0 0 | {3} | 0 0 -x_3 0 x_4 0 0 | {3} | 0 0 0 -x_4 0 x_5 0 | {3} | 0 0 0 0 -x_5 0 x_6 | {3} | x_7 0 0 0 0 -x_6 0 | a_3 = {4} | x_2x_4x_6 | {4} | x_3x_5x_7 | {4} | zx_4x_6 | {4} | x_2x_5x_7 | {4} | zx_3x_6 | {4} | x_2x_4x_7 | {4} | zx_3x_5 | ------------------------------------------------------------------------------------------------------------------------ res(J): b_1 = | x_2x_4 x_3x_5 zx_4 x_2x_5 zx_3 -zx_5 x_3x_6 x_4x_6 x_2x_6 | b_2 = {2} | 0 z 0 0 -x_5 0 0 0 0 0 0 x_6 0 0 0 0 | {2} | -z 0 x_2 0 0 -z 0 0 0 0 0 0 x_6 0 0 0 | {2} | 0 -x_2 0 x_3 0 0 -x_5 0 0 0 0 0 0 x_6 0 0 | {2} | 0 0 -x_3 0 x_4 0 0 -z 0 0 0 0 0 0 x_6 0 | {2} | x_5 0 0 -x_4 0 0 0 0 0 0 0 0 0 0 0 x_6 | {2} | 0 0 0 0 0 -x_3 -x_4 -x_2 0 0 0 0 0 0 0 0 | {2} | 0 0 0 0 0 0 0 0 0 -x_2 x_4 0 -x_5 0 0 -z | {2} | 0 0 0 0 0 0 0 0 x_2 0 -x_3 0 0 -z 0 0 | {2} | 0 0 0 0 0 0 0 0 -x_4 x_3 0 -x_4 0 0 -x_5 0 | b_3 = {3} | 0 0 -x_4 0 -x_6 0 0 0 0 | {3} | -x_5 0 0 0 0 -x_6 0 0 0 | {3} | 0 -z 0 0 0 0 -x_6 0 0 | {3} | 0 0 -x_5 0 0 0 0 -x_6 0 | {3} | -z 0 0 0 0 0 0 0 -x_6 | {3} | 0 -x_2 x_4 0 0 0 0 0 0 | {3} | x_2 0 -x_3 0 0 0 0 0 0 | {3} | -x_4 x_3 0 0 0 0 0 0 0 | {3} | 0 0 0 -x_3 0 -z 0 0 0 | {3} | 0 0 0 -x_4 0 0 -x_5 0 0 | {3} | 0 0 0 -x_2 0 0 0 -z 0 | {3} | 0 0 0 0 0 z 0 0 -x_5 | {3} | 0 0 0 0 -z 0 x_2 0 0 | {3} | 0 0 0 0 0 -x_2 0 x_3 0 | {3} | 0 0 0 0 0 0 -x_3 0 x_4 | {3} | 0 0 0 0 x_5 0 0 -x_4 0 | b_4 = {4} | -x_3x_6 | {4} | -x_4x_6 | {4} | -x_2x_6 | {4} | -zx_5 | {4} | x_2x_4 | {4} | x_3x_5 | {4} | zx_4 | {4} | x_2x_5 | {4} | zx_3 | ------------------------------------------------------------------------------------------------------------------------ phi: {x_2*x_4, x_3*x_5, z*x_4, x_2*x_5, z*x_3, -z*x_5, x_3*x_6, x_4*x_6, x_2*x_6} -> {0, 0, z*x_4*x_7, 0, z*x_3*x_7, -z*x_5*x_7, 0, 0, 0} ------------------------------------------------------------------------------------------------------------------------ alpha_1 = {2} | x_6 0 0 0 0 x_7 0 | {2} | 0 x_7 0 0 0 0 z | {2} | 0 0 0 0 0 0 0 | {2} | 0 0 0 x_7 0 0 0 | {2} | 0 0 0 0 0 0 0 | {2} | 0 0 0 0 0 0 0 | {2} | 0 0 0 0 z 0 0 | {2} | 0 0 z 0 0 0 0 | {2} | 0 0 0 0 0 0 0 | alpha_2 = {3} | 0 0 0 0 0 0 0 | {3} | 0 0 0 0 0 0 0 | {3} | 0 0 x_7 0 0 0 0 | {3} | 0 0 0 0 0 0 0 | {3} | 0 0 0 0 x_7 0 0 | {3} | 0 0 0 0 0 0 0 | {3} | 0 0 0 0 0 0 0 | {3} | 0 0 0 0 0 0 0 | {3} | 0 -z 0 0 0 0 0 | {3} | 0 0 0 0 0 0 0 | {3} | 0 0 0 -z 0 0 0 | {3} | 0 z 0 0 0 0 0 | {3} | 0 0 0 0 0 -z 0 | {3} | 0 0 0 0 0 0 0 | {3} | 0 0 0 0 0 0 0 | {3} | 0 0 0 0 0 0 0 | alpha_3 = {4} | 0 | {4} | 0 | {4} | 0 | {4} | zx_5x_7 | {4} | 0 | {4} | 0 | {4} | -zx_4x_7 | {4} | 0 | {4} | -zx_3x_7 | ------------------------------------------------------------------------------------------------------------------------ beta_1 = | 0 0 -zx_4x_7 0 -zx_3x_7 zx_5x_7 0 0 0 | beta_2 = {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | {3} | -z 0 0 0 0 -z 0 0 0 0 0 0 0 0 0 0 | {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 -x_7 0 0 | {3} | 0 0 0 0 0 0 0 -z 0 0 0 0 0 0 0 0 | {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -x_7 | {3} | 0 z 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | beta_3 = {4} | 0 0 0 0 -x_6 0 0 0 0 | {4} | 0 0 0 0 0 -x_7 0 0 0 | {4} | 0 z 0 0 0 0 0 0 0 | {4} | 0 0 0 0 0 0 0 -x_7 0 | {4} | z 0 0 0 0 0 0 0 0 | {4} | 0 0 0 0 -x_7 0 0 0 0 | {4} | 0 0 0 0 0 -z 0 0 0 | ------------------------------------------------------------------------------------------------------------------------ u = -1 ------------------------------------------------------------------------------------------------------------------------ h_1 = 0 h_2 = 0 h_3 = 0 ------------------------------------------------------------------------------------------------------------------------ f_1 = | x_2x_4x_6 x_3x_5x_7 zx_4x_6 x_2x_5x_7 zx_3x_6 x_2x_4x_7 zx_3x_5 x_8x_2x_4 x_8x_3x_5 x_8zx_4-zx_4x_7 x_8x_2x_5 x_8zx_3-zx_3x_7 -x_8zx_5+zx_5x_7 x_8x_3x_6 x_8x_4x_6 x_8x_2x_6 | f_2 = | 0 z 0 0 0 0 -x_7 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x_8 0 0 0 0 0 0 | | -z 0 x_2 0 0 0 0 -z 0 0 0 0 -z 0 0 0 0 0 0 0 0 0 0 0 x_8 0 0 0 0 0 | | 0 -x_2 0 x_3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -x_7 0 0 0 0 x_8 0 0 0 0 | | 0 0 -x_3 0 x_4 0 0 0 0 0 0 0 0 0 -z 0 0 0 0 0 0 0 0 0 0 0 x_8 0 0 0 | | 0 0 0 -x_4 0 x_5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -x_7 0 0 0 0 x_8 0 0 | | 0 0 0 0 -x_5 0 x_6 0 z 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x_8 0 | | x_7 0 0 0 0 -x_6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x_8 | | 0 0 0 0 0 0 0 0 -z 0 0 x_5 0 0 0 0 0 0 -x_6 0 0 0 0 -x_6 0 0 0 0 -x_7 0 | | 0 0 0 0 0 0 0 z 0 -x_2 0 0 z 0 0 0 0 0 0 -x_6 0 0 0 0 -x_7 0 0 0 0 -z | | 0 0 0 0 0 0 0 0 x_2 0 -x_3 0 0 x_5 0 0 0 0 0 0 -x_6 0 0 0 0 0 0 0 0 0 | | 0 0 0 0 0 0 0 0 0 x_3 0 -x_4 0 0 z 0 0 0 0 0 0 -x_6 0 0 0 0 -x_7 0 0 0 | | 0 0 0 0 0 0 0 -x_5 0 0 x_4 0 0 0 0 0 0 0 0 0 0 0 -x_6 0 0 0 0 0 0 0 | | 0 0 0 0 0 0 0 0 0 0 0 0 x_3 x_4 x_2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x_2 -x_4 0 x_5 0 0 z 0 0 0 0 -z 0 0 | | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -x_2 0 x_3 0 0 z 0 0 0 0 -z 0 0 0 0 | | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x_4 -x_3 0 x_4 0 0 x_5 0 0 0 0 0 0 0 0 | f_3 = | 0 0 0 0 -x_6 0 0 0 0 -x_8 0 0 0 0 0 0 | | 0 0 0 0 0 -x_7 0 0 0 0 -x_8 0 0 0 0 0 | | 0 z 0 0 0 0 0 0 0 0 0 -x_8 0 0 0 0 | | 0 0 0 0 0 0 0 -x_7 0 0 0 0 -x_8 0 0 0 | | z 0 0 0 0 0 0 0 0 0 0 0 0 -x_8 0 0 | | 0 0 0 0 -x_7 0 0 0 0 0 0 0 0 0 -x_8 0 | | 0 0 0 0 0 -z 0 0 0 0 0 0 0 0 0 -x_8 | | 0 0 x_4 0 x_6 0 0 0 0 0 0 0 0 0 0 0 | | x_5 0 0 0 0 x_6 0 0 0 0 0 0 0 0 0 0 | | 0 z 0 0 0 0 x_6 0 0 0 0 -x_7 0 0 0 0 | | 0 0 x_5 0 0 0 0 x_6 0 0 0 0 0 0 0 0 | | z 0 0 0 0 0 0 0 x_6 0 0 0 0 -x_7 0 0 | | 0 x_2 -x_4 0 0 0 0 0 0 0 0 0 0 0 0 0 | | -x_2 0 x_3 0 0 0 0 0 0 0 0 0 0 0 0 0 | | x_4 -x_3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | | 0 0 0 x_3 0 z 0 0 0 0 z 0 0 0 0 0 | | 0 0 0 x_4 0 0 x_5 0 0 0 0 0 0 0 0 0 | | 0 0 0 x_2 0 0 0 z 0 0 0 0 z 0 0 0 | | 0 0 0 0 0 -z 0 0 x_5 0 -z 0 0 0 0 0 | | 0 0 0 0 z 0 -x_2 0 0 0 0 0 0 0 z 0 | | 0 0 0 0 0 x_2 0 -x_3 0 0 0 0 0 0 0 0 | | 0 0 0 0 0 0 x_3 0 -x_4 0 0 0 0 0 0 0 | | 0 0 0 0 -x_5 0 0 x_4 0 0 0 0 0 0 0 0 | | 0 0 0 0 0 0 0 0 0 0 z 0 0 0 0 -x_7 | | 0 0 0 0 0 0 0 0 0 -z 0 x_2 0 0 0 0 | | 0 0 0 0 0 0 0 0 0 0 -x_2 0 x_3 0 0 0 | | 0 0 0 0 0 0 0 0 0 0 0 -x_3 0 x_4 0 0 | | 0 0 0 0 0 0 0 0 0 0 0 0 -x_4 0 x_5 0 | | 0 0 0 0 0 0 0 0 0 0 0 0 0 -x_5 0 x_6 | | 0 0 0 0 0 0 0 0 0 x_7 0 0 0 0 -x_6 0 | f_4 = | x_8x_3x_6 | | x_8x_4x_6 | | x_8x_2x_6 | | x_8zx_5-zx_5x_7 | | -x_8x_2x_4 | | -x_8x_3x_5 | | -x_8zx_4+zx_4x_7 | | -x_8x_2x_5 | | -x_8zx_3+zx_3x_7 | | x_2x_4x_6 | | x_3x_5x_7 | | zx_4x_6 | | x_2x_5x_7 | | zx_3x_6 | | x_2x_4x_7 | | zx_3x_5 | ------------------------------------------------------------------------------------------------------------------------ ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ delta(6,{x_1, x_2, x_3, x_4, x_5, x_6, x_7, x_8, x_9}) + delta(4,{z, x_2, x_3, x_4, x_5, x_6, x_7, x_8}) -> delta(6,{x_1, x_2, x_3, x_4, x_5, x_6, x_7, x_8, x_9, x_10}) ------------------------------------------------------------------------------------------------------------------------ res(I): a_1 = | x_2x_4x_6x_8 x_3x_5x_7x_9 x_1x_4x_6x_8 x_2x_5x_7x_9 x_1x_3x_6x_8 x_2x_4x_7x_9 x_1x_3x_5x_8 x_2x_4x_6x_9 x_1x_3x_5x_7 | a_2 = {4} | 0 x_1 0 0 0 0 0 0 -x_9 | {4} | -x_1 0 x_2 0 0 0 0 0 0 | {4} | 0 -x_2 0 x_3 0 0 0 0 0 | {4} | 0 0 -x_3 0 x_4 0 0 0 0 | {4} | 0 0 0 -x_4 0 x_5 0 0 0 | {4} | 0 0 0 0 -x_5 0 x_6 0 0 | {4} | 0 0 0 0 0 -x_6 0 x_7 0 | {4} | 0 0 0 0 0 0 -x_7 0 x_8 | {4} | x_9 0 0 0 0 0 0 -x_8 0 | a_3 = {5} | x_2x_4x_6x_8 | {5} | x_3x_5x_7x_9 | {5} | x_1x_4x_6x_8 | {5} | x_2x_5x_7x_9 | {5} | x_1x_3x_6x_8 | {5} | x_2x_4x_7x_9 | {5} | x_1x_3x_5x_8 | {5} | x_2x_4x_6x_9 | {5} | x_1x_3x_5x_7 | ------------------------------------------------------------------------------------------------------------------------ res(J): b_1 = | x_2x_4x_6 x_3x_5x_7 zx_4x_6 x_2x_5x_7 zx_3x_6 x_2x_4x_7 zx_3x_5 x_2x_4x_8 x_3x_5x_8 -zx_4x_7 x_2x_5x_8 -zx_3x_7 zx_5x_7 x_3x_6x_8 x_4x_6x_8 x_2x_6x_8 | b_2 = {3} | 0 z 0 0 0 0 -x_7 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x_8 0 0 0 0 0 0 | {3} | -z 0 x_2 0 0 0 0 -z 0 0 0 0 -z 0 0 0 0 0 0 0 0 0 0 0 x_8 0 0 0 0 0 | {3} | 0 -x_2 0 x_3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -x_7 0 0 0 0 x_8 0 0 0 0 | {3} | 0 0 -x_3 0 x_4 0 0 0 0 0 0 0 0 0 -z 0 0 0 0 0 0 0 0 0 0 0 x_8 0 0 0 | {3} | 0 0 0 -x_4 0 x_5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -x_7 0 0 0 0 x_8 0 0 | {3} | 0 0 0 0 -x_5 0 x_6 0 z 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x_8 0 | {3} | x_7 0 0 0 0 -x_6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x_8 | {3} | 0 0 0 0 0 0 0 0 0 0 0 x_5 0 0 0 0 0 0 -x_6 0 0 0 0 -x_6 0 0 0 0 -x_7 0 | {3} | 0 0 0 0 0 0 0 0 0 -x_2 0 0 0 0 0 0 0 0 0 -x_6 0 0 0 0 -x_7 0 0 0 0 -z | {3} | 0 0 0 0 0 0 0 0 x_2 0 -x_3 0 0 x_5 0 0 0 0 0 0 -x_6 0 0 0 0 0 0 0 0 0 | {3} | 0 0 0 0 0 0 0 0 0 x_3 0 -x_4 0 0 0 0 0 0 0 0 0 -x_6 0 0 0 0 -x_7 0 0 0 | {3} | 0 0 0 0 0 0 0 -x_5 0 0 x_4 0 0 0 0 0 0 0 0 0 0 0 -x_6 0 0 0 0 0 0 0 | {3} | 0 0 0 0 0 0 0 0 0 0 0 0 x_3 x_4 x_2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x_2 -x_4 0 x_5 0 0 0 0 0 0 0 -z 0 0 | {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -x_2 0 x_3 0 0 0 0 0 0 0 -z 0 0 0 0 | {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x_4 -x_3 0 x_4 0 0 x_5 0 0 0 0 0 0 0 0 | b_3 = {4} | 0 0 0 0 -x_6 0 0 0 0 -x_8 0 0 0 0 0 0 | {4} | 0 0 0 0 0 -x_7 0 0 0 0 -x_8 0 0 0 0 0 | {4} | 0 z 0 0 0 0 0 0 0 0 0 -x_8 0 0 0 0 | {4} | 0 0 0 0 0 0 0 -x_7 0 0 0 0 -x_8 0 0 0 | {4} | z 0 0 0 0 0 0 0 0 0 0 0 0 -x_8 0 0 | {4} | 0 0 0 0 -x_7 0 0 0 0 0 0 0 0 0 -x_8 0 | {4} | 0 0 0 0 0 -z 0 0 0 0 0 0 0 0 0 -x_8 | {4} | 0 0 x_4 0 x_6 0 0 0 0 0 0 0 0 0 0 0 | {4} | x_5 0 0 0 0 x_6 0 0 0 0 0 0 0 0 0 0 | {4} | 0 0 0 0 0 0 x_6 0 0 0 0 -x_7 0 0 0 0 | {4} | 0 0 x_5 0 0 0 0 x_6 0 0 0 0 0 0 0 0 | {4} | 0 0 0 0 0 0 0 0 x_6 0 0 0 0 -x_7 0 0 | {4} | 0 x_2 -x_4 0 0 0 0 0 0 0 0 0 0 0 0 0 | {4} | -x_2 0 x_3 0 0 0 0 0 0 0 0 0 0 0 0 0 | {4} | x_4 -x_3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | {4} | 0 0 0 x_3 0 0 0 0 0 0 z 0 0 0 0 0 | {4} | 0 0 0 x_4 0 0 x_5 0 0 0 0 0 0 0 0 0 | {4} | 0 0 0 x_2 0 0 0 0 0 0 0 0 z 0 0 0 | {4} | 0 0 0 0 0 0 0 0 x_5 0 -z 0 0 0 0 0 | {4} | 0 0 0 0 0 0 -x_2 0 0 0 0 0 0 0 z 0 | {4} | 0 0 0 0 0 x_2 0 -x_3 0 0 0 0 0 0 0 0 | {4} | 0 0 0 0 0 0 x_3 0 -x_4 0 0 0 0 0 0 0 | {4} | 0 0 0 0 -x_5 0 0 x_4 0 0 0 0 0 0 0 0 | {4} | 0 0 0 0 0 0 0 0 0 0 z 0 0 0 0 -x_7 | {4} | 0 0 0 0 0 0 0 0 0 -z 0 x_2 0 0 0 0 | {4} | 0 0 0 0 0 0 0 0 0 0 -x_2 0 x_3 0 0 0 | {4} | 0 0 0 0 0 0 0 0 0 0 0 -x_3 0 x_4 0 0 | {4} | 0 0 0 0 0 0 0 0 0 0 0 0 -x_4 0 x_5 0 | {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 -x_5 0 x_6 | {4} | 0 0 0 0 0 0 0 0 0 x_7 0 0 0 0 -x_6 0 | b_4 = {5} | x_3x_6x_8 | {5} | x_4x_6x_8 | {5} | x_2x_6x_8 | {5} | -zx_5x_7 | {5} | -x_2x_4x_8 | {5} | -x_3x_5x_8 | {5} | zx_4x_7 | {5} | -x_2x_5x_8 | {5} | zx_3x_7 | {5} | x_2x_4x_6 | {5} | x_3x_5x_7 | {5} | zx_4x_6 | {5} | x_2x_5x_7 | {5} | zx_3x_6 | {5} | x_2x_4x_7 | {5} | zx_3x_5 | ------------------------------------------------------------------------------------------------------------------------ phi: {x_2*x_4*x_6, x_3*x_5*x_7, z*x_4*x_6, x_2*x_5*x_7, z*x_3*x_6, x_2*x_4*x_7, z*x_3*x_5, x_2*x_4*x_8, x_3*x_5*x_8, -z*x_4*x_7, x_2*x_5*x_8, -z*x_3*x_7, z*x_5*x_7, x_3*x_6*x_8, x_4*x_6*x_8, x_2*x_6*x_8} -> {0, 0, x_1*x_4*x_6*x_9, 0, x_1*x_3*x_6*x_9, 0, x_1*x_3*x_5*x_9, 0, 0, -x_1*x_4*x_7*x_9, 0, -x_1*x_3*x_7*x_9, x_1*x_5*x_7*x_9, 0, 0, 0} ------------------------------------------------------------------------------------------------------------------------ alpha_1 = {3} | x_8 0 0 0 0 0 0 x_9 0 | {3} | 0 x_9 0 0 0 0 0 0 x_1 | {3} | 0 0 0 0 0 0 0 0 0 | {3} | 0 0 0 x_9 0 0 0 0 0 | {3} | 0 0 0 0 0 0 0 0 0 | {3} | 0 0 0 0 0 x_9 0 0 0 | {3} | 0 0 0 0 0 0 0 0 0 | {3} | 0 0 0 0 0 0 0 0 0 | {3} | 0 0 0 0 0 0 x_1 0 0 | {3} | 0 0 0 0 0 0 0 0 0 | {3} | 0 0 0 0 0 0 0 0 0 | {3} | 0 0 0 0 0 0 0 0 0 | {3} | 0 0 0 0 0 0 0 0 0 | {3} | 0 0 0 0 x_1 0 0 0 0 | {3} | 0 0 x_1 0 0 0 0 0 0 | {3} | 0 0 0 0 0 0 0 0 0 | alpha_2 = {4} | 0 0 0 0 0 0 0 0 0 | {4} | 0 0 0 0 0 0 0 0 0 | {4} | 0 0 x_9 0 0 0 0 0 0 | {4} | 0 0 0 0 0 0 0 0 0 | {4} | 0 0 0 0 x_9 0 0 0 0 | {4} | 0 0 0 0 0 0 0 0 0 | {4} | 0 0 0 0 0 0 x_9 0 0 | {4} | 0 0 0 0 0 0 0 0 0 | {4} | 0 0 0 0 0 0 0 0 0 | {4} | 0 0 0 0 0 0 0 0 0 | {4} | 0 0 0 0 0 0 0 0 0 | {4} | 0 0 0 0 0 0 0 0 0 | {4} | 0 0 0 0 0 0 0 0 0 | {4} | 0 0 0 0 0 0 0 0 0 | {4} | 0 0 0 0 0 0 0 0 0 | {4} | 0 x_1 0 0 0 0 0 0 0 | {4} | 0 0 0 0 0 0 0 0 0 | {4} | 0 0 0 x_1 0 0 0 0 0 | {4} | 0 -x_1 0 0 0 0 0 0 0 | {4} | 0 0 0 0 0 x_1 0 0 0 | {4} | 0 0 0 0 0 0 0 0 0 | {4} | 0 0 0 0 0 0 0 0 0 | {4} | 0 0 0 0 0 0 0 0 0 | {4} | 0 x_1 0 0 0 0 0 0 0 | {4} | 0 0 0 0 0 0 0 -x_1 0 | {4} | 0 0 0 0 0 0 0 0 0 | {4} | 0 0 0 0 0 0 0 0 0 | {4} | 0 0 0 0 0 0 0 0 0 | {4} | 0 0 0 0 0 0 0 0 0 | {4} | 0 0 0 0 0 0 0 0 0 | alpha_3 = {5} | 0 | {5} | 0 | {5} | 0 | {5} | x_1x_5x_7x_9 | {5} | 0 | {5} | 0 | {5} | -x_1x_4x_7x_9 | {5} | 0 | {5} | -x_1x_3x_7x_9 | {5} | 0 | {5} | 0 | {5} | -x_1x_4x_6x_9 | {5} | 0 | {5} | -x_1x_3x_6x_9 | {5} | 0 | {5} | -x_1x_3x_5x_9 | ------------------------------------------------------------------------------------------------------------------------ beta_1 = | 0 0 -x_1x_4x_6x_9 0 -x_1x_3x_6x_9 0 -x_1x_3x_5x_9 0 0 x_1x_4x_7x_9 0 x_1x_3x_7x_9 -x_1x_5x_7x_9 0 0 0 | beta_2 = {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | {4} | -x_1 0 0 0 0 0 0 -x_1 0 0 0 0 -x_1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -x_9 0 0 0 0 | {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -x_1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -x_9 0 0 | {4} | 0 0 0 0 0 0 0 0 x_1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -x_9 | {4} | 0 x_1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | beta_3 = {5} | 0 0 0 0 0 0 0 0 0 -x_8 0 0 0 0 0 0 | {5} | 0 0 0 0 0 0 0 0 0 0 -x_9 0 0 0 0 0 | {5} | 0 -x_1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | {5} | 0 0 0 0 0 0 0 0 0 0 0 0 -x_9 0 0 0 | {5} | -x_1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -x_9 0 | {5} | 0 0 0 0 0 x_1 0 0 0 0 0 0 0 0 0 0 | {5} | 0 0 0 0 0 0 0 0 0 -x_9 0 0 0 0 0 0 | {5} | 0 0 0 0 0 0 0 0 0 0 -x_1 0 0 0 0 0 | ------------------------------------------------------------------------------------------------------------------------ u = -1 ------------------------------------------------------------------------------------------------------------------------ h_1 = 0 h_2 = 0 h_3 = 0 ------------------------------------------------------------------------------------------------------------------------ f_1 = | x_2x_4x_6x_8 x_3x_5x_7x_9 x_1x_4x_6x_8 x_2x_5x_7x_9 x_1x_3x_6x_8 x_2x_4x_7x_9 x_1x_3x_5x_8 x_2x_4x_6x_9 x_1x_3x_5x_7 x_10x_2x_4x_6 x_10x_3x_5x_7 x_10zx_4x_6-x_1x_4x_6x_9 x_10x_2x_5x_7 x_10zx_3x_6-x_1x_3x_6x_9 x_10x_2x_4x_7 x_10zx_3x_5-x_1x_3x_5x_9 x_10x_2x_4x_8 x_10x_3x_5x_8 -x_10zx_4x_7+x_1x_4x_7x_9 x_10x_2x_5x_8 -x_10zx_3x_7+x_1x_3x_7x_9 x_10zx_5x_7-x_1x_5x_7x_9 x_10x_3x_6x_8 x_10x_4x_6x_8 x_10x_2x_6x_8 | f_2 = | 0 x_1 0 0 0 0 0 0 -x_9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x_10 0 0 0 0 0 0 0 0 | | -x_1 0 x_2 0 0 0 0 0 0 -x_1 0 0 0 0 0 0 -x_1 0 0 0 0 -x_1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x_10 0 0 0 0 0 0 0 | | 0 -x_2 0 x_3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -x_9 0 0 0 0 0 0 x_10 0 0 0 0 0 0 | | 0 0 -x_3 0 x_4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -x_1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x_10 0 0 0 0 0 | | 0 0 0 -x_4 0 x_5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -x_9 0 0 0 0 0 0 x_10 0 0 0 0 | | 0 0 0 0 -x_5 0 x_6 0 0 0 0 0 0 0 0 0 0 x_1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x_10 0 0 0 | | 0 0 0 0 0 -x_6 0 x_7 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -x_9 0 0 0 0 0 0 x_10 0 0 | | 0 0 0 0 0 0 -x_7 0 x_8 0 x_1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x_10 0 | | x_9 0 0 0 0 0 0 -x_8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x_10 | | 0 0 0 0 0 0 0 0 0 0 -z 0 0 0 0 x_7 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -x_8 0 0 0 0 0 0 -x_8 0 0 0 0 0 0 -x_9 0 | | 0 0 0 0 0 0 0 0 0 z 0 -x_2 0 0 0 0 z 0 0 0 0 z 0 0 0 0 0 0 0 0 0 0 0 -x_8 0 0 0 0 0 0 -x_9 0 0 0 0 0 0 -x_1 | | 0 0 0 0 0 0 0 0 0 0 x_2 0 -x_3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x_7 0 0 0 0 -x_8 0 0 0 0 0 0 0 0 0 0 0 0 0 | | 0 0 0 0 0 0 0 0 0 0 0 x_3 0 -x_4 0 0 0 0 0 0 0 0 0 z 0 0 0 0 0 0 0 0 0 0 0 -x_8 0 0 0 0 0 0 -x_9 0 0 0 0 0 | | 0 0 0 0 0 0 0 0 0 0 0 0 x_4 0 -x_5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x_7 0 0 0 0 -x_8 0 0 0 0 0 0 0 0 0 0 0 | | 0 0 0 0 0 0 0 0 0 0 0 0 0 x_5 0 -x_6 0 -z 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -x_8 0 0 0 0 0 0 -x_9 0 0 0 | | 0 0 0 0 0 0 0 0 0 -x_7 0 0 0 0 x_6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -x_8 0 0 0 0 0 0 0 0 0 | | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -x_5 0 0 0 0 0 0 x_6 0 0 0 0 x_6 0 0 0 0 x_7 0 0 0 0 0 0 0 0 0 0 | | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x_2 0 0 0 0 0 0 0 0 0 x_6 0 0 0 0 x_7 0 0 0 0 z 0 0 0 0 0 0 -x_1 0 0 | | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -x_2 0 x_3 0 0 -x_5 0 0 0 0 0 0 x_6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -x_3 0 x_4 0 0 0 0 0 0 0 0 0 x_6 0 0 0 0 x_7 0 0 0 0 0 0 0 0 0 0 0 0 | | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x_5 0 0 -x_4 0 0 0 0 0 0 0 0 0 0 0 x_6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -x_3 -x_4 -x_2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -x_2 x_4 0 -x_5 0 0 0 0 0 0 0 z 0 0 0 0 0 0 -x_1 0 0 0 0 | | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x_2 0 -x_3 0 0 0 0 0 0 0 z 0 0 0 0 0 0 -x_1 0 0 0 0 0 0 | | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -x_4 x_3 0 -x_4 0 0 -x_5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | f_3 = | 0 0 0 0 0 0 0 0 0 -x_8 0 0 0 0 0 0 -x_10 0 0 0 0 0 0 0 0 | | 0 0 0 0 0 0 0 0 0 0 -x_9 0 0 0 0 0 0 -x_10 0 0 0 0 0 0 0 | | 0 -x_1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -x_10 0 0 0 0 0 0 | | 0 0 0 0 0 0 0 0 0 0 0 0 -x_9 0 0 0 0 0 0 -x_10 0 0 0 0 0 | | -x_1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -x_10 0 0 0 0 | | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -x_9 0 0 0 0 0 0 -x_10 0 0 0 | | 0 0 0 0 0 x_1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -x_10 0 0 | | 0 0 0 0 0 0 0 0 0 -x_9 0 0 0 0 0 0 0 0 0 0 0 0 0 -x_10 0 | | 0 0 0 0 0 0 0 0 0 0 -x_1 0 0 0 0 0 0 0 0 0 0 0 0 0 -x_10 | | 0 0 0 0 x_6 0 0 0 0 x_8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | | 0 0 0 0 0 x_7 0 0 0 0 x_8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | | 0 -z 0 0 0 0 0 0 0 0 0 x_8 0 0 0 0 0 0 -x_9 0 0 0 0 0 0 | | 0 0 0 0 0 0 0 x_7 0 0 0 0 x_8 0 0 0 0 0 0 0 0 0 0 0 0 | | -z 0 0 0 0 0 0 0 0 0 0 0 0 x_8 0 0 0 0 0 0 -x_9 0 0 0 0 | | 0 0 0 0 x_7 0 0 0 0 0 0 0 0 0 x_8 0 0 0 0 0 0 0 0 0 0 | | 0 0 0 0 0 z 0 0 0 0 0 0 0 0 0 x_8 0 0 0 0 0 0 -x_9 0 0 | | 0 0 -x_4 0 -x_6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | | -x_5 0 0 0 0 -x_6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | | 0 0 0 0 0 0 -x_6 0 0 0 0 x_7 0 0 0 0 0 0 0 0 0 0 0 0 0 | | 0 0 -x_5 0 0 0 0 -x_6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | | 0 0 0 0 0 0 0 0 -x_6 0 0 0 0 x_7 0 0 0 0 0 0 0 0 0 0 0 | | 0 -x_2 x_4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | | x_2 0 -x_3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | | -x_4 x_3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | | 0 0 0 -x_3 0 0 0 0 0 0 -z 0 0 0 0 0 0 -x_1 0 0 0 0 0 0 0 | | 0 0 0 -x_4 0 0 -x_5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | | 0 0 0 -x_2 0 0 0 0 0 0 0 0 -z 0 0 0 0 0 0 -x_1 0 0 0 0 0 | | 0 0 0 0 0 0 0 0 -x_5 0 z 0 0 0 0 0 0 x_1 0 0 0 0 0 0 0 | | 0 0 0 0 0 0 x_2 0 0 0 0 0 0 0 -z 0 0 0 0 0 0 -x_1 0 0 0 | | 0 0 0 0 0 -x_2 0 x_3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | | 0 0 0 0 0 0 -x_3 0 x_4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | | 0 0 0 0 x_5 0 0 -x_4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | | 0 0 0 0 0 0 0 0 0 0 -z 0 0 0 0 x_7 0 -x_1 0 0 0 0 0 0 0 | | 0 0 0 0 0 0 0 0 0 z 0 -x_2 0 0 0 0 0 0 0 0 0 0 0 x_1 0 | | 0 0 0 0 0 0 0 0 0 0 x_2 0 -x_3 0 0 0 0 0 0 0 0 0 0 0 0 | | 0 0 0 0 0 0 0 0 0 0 0 x_3 0 -x_4 0 0 0 0 0 0 0 0 0 0 0 | | 0 0 0 0 0 0 0 0 0 0 0 0 x_4 0 -x_5 0 0 0 0 0 0 0 0 0 0 | | 0 0 0 0 0 0 0 0 0 0 0 0 0 x_5 0 -x_6 0 0 0 0 0 0 0 0 0 | | 0 0 0 0 0 0 0 0 0 -x_7 0 0 0 0 x_6 0 0 0 0 0 0 0 0 0 0 | | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x_1 0 0 0 0 0 0 -x_9 | | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -x_1 0 x_2 0 0 0 0 0 0 | | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -x_2 0 x_3 0 0 0 0 0 | | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -x_3 0 x_4 0 0 0 0 | | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -x_4 0 x_5 0 0 0 | | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -x_5 0 x_6 0 0 | | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -x_6 0 x_7 0 | | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -x_7 0 x_8 | | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x_9 0 0 0 0 0 0 -x_8 0 | f_4 = | -x_10x_3x_6x_8 | | -x_10x_4x_6x_8 | | -x_10x_2x_6x_8 | | x_10zx_5x_7-x_1x_5x_7x_9 | | x_10x_2x_4x_8 | | x_10x_3x_5x_8 | | -x_10zx_4x_7+x_1x_4x_7x_9 | | x_10x_2x_5x_8 | | -x_10zx_3x_7+x_1x_3x_7x_9 | | -x_10x_2x_4x_6 | | -x_10x_3x_5x_7 | | -x_10zx_4x_6+x_1x_4x_6x_9 | | -x_10x_2x_5x_7 | | -x_10zx_3x_6+x_1x_3x_6x_9 | | -x_10x_2x_4x_7 | | -x_10zx_3x_5+x_1x_3x_5x_9 | | x_2x_4x_6x_8 | | x_3x_5x_7x_9 | | x_1x_4x_6x_8 | | x_2x_5x_7x_9 | | x_1x_3x_6x_8 | | x_2x_4x_7x_9 | | x_1x_3x_5x_8 | | x_2x_4x_6x_9 | | x_1x_3x_5x_7 | ------------------------------------------------------------------------------------------------------------------------```