############## #This program attempts to calculates the matrix #corresponding to the map d_3 in a resolution of the trivial #module k as in Lemma 3.4, for any of the examples A-G given in #Section 3.3. Below, we only feed the program the solutions #corresponding to examples D(1,h), E(1, \gamma), F(1, \gamma), #G(1, \gamma), since these are the only cases in which #calculating the resolution was necessary to prove the algebra #is AS-regular. ############################ with(LinearAlgebra); sol_sub_seq:=[{c = 1, q = -1, p = -1, d = 0, e = 0, g = 0, k = 0, j = 0, h = h, a = -1, f = -1+h, b = 1}, {q = -1, h = 0, p = -1, f = 0, e = 0, c = 0, j = 1, d = 1, g = RootOf(_Z^2+1), k = 1, a = -1, b = 1}, {e = -1, h = 0, f = 0, j = 0, c = 0, q = RootOf(_Z^2+_Z+1), p = -1-RootOf(_Z^2+_Z+1), a = -RootOf(_Z^2+_Z+1), d = -1-RootOf(_Z^2+_Z+1), b = 1, g = RootOf(_Z^2+_Z+1), k = -1-RootOf(_Z^2+_Z+1)}, {q = -1, p = -1, e = 0, g = 0, k = 0, j = 0, c = 0, d = 1, a = -1, b = 1, h = RootOf(2*_Z^2-2*_Z+1), f = 1-RootOf(2*_Z^2-2*_Z+1)}]; rr1d2_sub_list:= {x[2,1]= p*x[1,2]}; rr2d2_sub_list:={x[3,2] = a*x[3,1] + x[1,3] + b*x[2,3]}; rr1d3_sub_list:= {seq(x[r,2,1]= p*x[r,1,2], r = 1..3)} union {seq(x[2,1,s]= p*x[1,2,s], s = 1..3)}; rr2d3_sub_list:={seq(x[r,3,2] = a*x[r,3,1] + x[r,1,3] + b*x[r,2,3], r =1..3)} union {seq(x[3,2,s] = a*x[3,1,s] + x[1,3,s] + b*x[2,3,s], s = 1..3)}; rr3d3_sub_list:={x[3,3,1] = c*x[1,3,3] + d*x[2,3,3]+e*x[3,1,3]}; rr4d3_sub_list:= {x[3,1,1] = f*x[1,1,3] + g*x[1,2,3] +h*x[2,2,3] + j*x[1,3,1] + k*x[2,3,1]}; rr5d3_sub_list:= {x[3,1,2] = q*x[1,3,1] + q*b*x[2,3,1] + a*q*x[3,1,1]}; rr1d4_sub_list:= {seq(seq(x[r,s,2,1]= p*x[r,s,1,2], r = 1..3), s =1..3)} union {seq(seq(x[r,2,1,s]= p*x[r,1,2,s], r = 1..3), s =1..3)} union {seq(seq(x[2,1,r,s]= p*x[1,2,r,s], r = 1..3), s =1..3)}; rr2d4_sub_list:={seq(seq(x[r,s,3,2] = a*x[r,s,3,1] +x[r,s,1,3] + b*x[r,s,2,3], r =1..3), s = 1..3)} union {seq(seq(x[r,3,2,s] = a*x[r,3,1,s] + x[r,1,3,s] + b*x[r,2,3,s], r =1..3), s = 1..3)} union {seq(seq(x[3,2,r,s] = a*x[3,1,r,s] +x[1,3,r,s] + b*x[2,3,r,s], r = 1..3), s = 1..3)}; rr3d4_sub_list:={seq(x[r,3,3,1] = c*x[r,1,3,3] +d*x[r,2,3,3]+e*x[r,3,1,3], r = 1..3)} union {seq(x[3,3,1,r] = c*x[1,3,3,r] +d*x[2,3,3,r]+ e*x[3,1,3,r], r = 1..3)}; rr4d4_sub_list:= {seq(x[r,3,1,1] = f*x[r,1,1,3] + g*x[r,1,2,3] + h*x[r,2,2,3] + j*x[r,1,3,1] + k*x[r,2,3,1], r = 1..3)} union {seq(x[3,1,1,r] = f*x[1,1,3,r] +g*x[1,2,3,r] + h*x[2,2,3,r] +j*x[1,3,1,r] + k*x[2,3,1,r], r = 1..3)}; rr5d4_sub_list:= {seq(x[r,3,1,2] = q*x[r,1,3,1] + q*b*x[r,2,3,1] +a*q*x[r,3,1,1], r = 1..3)} union {seq(x[3,1,2,r] = q*x[1,3,1,r] + q*b*x[2,3,1,r] + a*q*x[3,1,1,r], r = 1..3)}; rr1d5_sub_list:= {seq(seq(seq(x[r,s,t,2,1]= p*x[r,s,t,1,2], r = 1..3), s = 1..3), t= 1..3)} union {seq(seq(seq(x[r,s,2,1,t]= p*x[r,s,1,2,t], r = 1..3), s = 1..3), t= 1..3)} union {seq(seq(seq(x[r,2,1,s,t]= p*x[r,1,2,s,t], r = 1..3), s = 1..3), t= 1..3)} union {seq(seq(seq(x[2,1,r,s,t]= p*x[1,2,r,s,t], r = 1..3), s = 1..3), t= 1..3)}; rr2d5_sub_list:= {seq(seq(seq(x[r,s,t,3,2]= a*x[r,s,t,3,1] +x[r,s,t,1,3]+b*x[r,s,t,2,3], r = 1..3), s = 1..3), t= 1..3)} union {seq(seq(seq(x[r,s,3,2,t]=a*x[r,s,3,1,t] +x[r,s,1,3,t]+b*x[r,s,2,3,t], r = 1..3), s = 1..3), t= 1..3)} union {seq(seq(seq(x[r,3,2,s,t]=a*x[r,3,1,s,t] +x[r,1,3,s,t]+b*x[r,2,3,s,t], r = 1..3), s = 1..3), t= 1..3)} union {seq(seq(seq(x[3,2,r,s,t]=a*x[3,1,r,s,t] +x[1,3,r,s,t]+b*x[2,3,r,s,t], r = 1..3), s = 1..3), t= 1..3)}; rr3d5_sub_list:= {seq(seq(x[r,s,3,3,1]= c*x[r,s,1,3,3] +d*x[r,s,2,3,3]+e*x[r,s,3,1,3], r = 1..3), s = 1..3)} union {seq(seq(x[r,3,3,1,s]= c*x[r,1,3,3,s] +d*x[r,2,3,3,s] +e*x[r,3,1,3,s], r = 1..3), s = 1..3)} union {seq(seq(x[3,3,1,r,s]= c*x[1,3,3,r,s] +d*x[2,3,3,r,s] +e*x[3,1,3,r,s], r = 1..3), s = 1..3)}; rr4d5_sub_list:= {seq(seq(x[r,s,3,1,1]= f*x[r,s,1,1,3] +g*x[r,s,1,2,3]+h*x[r,s,2,2,3]+j*x[r,s,1,3,1]+k*[r,s,2,3,1], r = 1..3), s = 1..3)} union {seq(seq(x[r,3,1,1,s]= f*x[r,1,1,3,s] +g*x[r,1,2,3,s] +h*x[r,2,2,3,s]+j*x[r,1,3,1,s]+k*[r,2,3,1,2], r = 1..3), s = 1..3)} union {seq(seq(x[3,1,1,r,s]= f*x[1,1,3,r,s] +g*x[1,2,3,r,s] +h*x[2,2,3,r,s]+j*x[1,3,1,r,s]+k*[2,3,1,r,s], r = 1..3), s = 1..3)}; rr5d5_sub_list:= {seq(seq(x[r,s,3,1,2]= q*x[r,s,1,3,1] +q*b*x[r,s,2,3,1]+a*q*x[r,s,3,1,1], r = 1..3), s = 1..3)} union {seq(seq(x[r,3,1,2,s]= q*x[r,1,3,1,s] +q*b*x[r,2,3,1,s]+a*q*x[r,3,1,1,s], r = 1..3), s = 1..3)} union {seq(seq(x[3,1,2,r,s]= q*x[1,3,1,r,s] +q*b*x[2,3,1,r,s]+a*q*x[3,1,1,r,s], r = 1..3), s = 1..3)}; for m from 1 by 1 to nops(sol_sub_seq) do r1d2_sub_list[m]:=subs(sol_sub_seq[m], rr1d2_sub_list); r2d2_sub_list[m]:=subs(sol_sub_seq[m], rr2d2_sub_list); r1d3_sub_list[m]:=subs(sol_sub_seq[m], rr1d3_sub_list); r2d3_sub_list[m]:=subs(sol_sub_seq[m], rr2d3_sub_list); r3d3_sub_list[m]:=subs(sol_sub_seq[m], rr3d3_sub_list); r4d3_sub_list[m]:=subs(sol_sub_seq[m], rr4d3_sub_list); r5d3_sub_list[m]:=subs(sol_sub_seq[m], rr5d3_sub_list); r1d4_sub_list[m]:=subs(sol_sub_seq[m], rr1d4_sub_list); r2d4_sub_list[m]:=subs(sol_sub_seq[m], rr2d4_sub_list); r3d4_sub_list[m]:=subs(sol_sub_seq[m], rr3d4_sub_list); r4d4_sub_list[m]:=subs(sol_sub_seq[m], rr4d4_sub_list); r5d4_sub_list[m]:=subs(sol_sub_seq[m], rr5d4_sub_list); r1d5_sub_list[m]:=subs(sol_sub_seq[m], rr1d5_sub_list); r2d5_sub_list[m]:=subs(sol_sub_seq[m], rr2d5_sub_list); r3d5_sub_list[m]:=subs(sol_sub_seq[m], rr3d5_sub_list); r4d5_sub_list[m]:=subs(sol_sub_seq[m], rr4d5_sub_list); r5d5_sub_list[m]:=subs(sol_sub_seq[m], rr5d5_sub_list); M[m,1,0]:=subs(sol_sub_seq[m], Matrix([[0, 0 ,0 ,0 ], [-1, 0, 0, 0], [0, a, 0, 0]])); M[m,2,0]:=subs(sol_sub_seq[m],Matrix([[p, 0 ,0, 0], [0, 0, 0, 0], [0,-1, 0, 0]])); M[m,3,0]:=subs(sol_sub_seq[m],Matrix([[0, 1, 0, 0], [0, b, 0, 0], [0, 0, 0, 0]])); M[m,1,1]:=subs(sol_sub_seq[m],Matrix([[0 , 0,0 , 0], [ 0,0 ,0 ,0 ], [0 , 0, 0,-1 ]])); M[m,1,2]:=subs(sol_sub_seq[m],Matrix([[ 0,0 ,0 ,0 ], [0 ,0 ,0 ,0 ], [ 0, 0, 0, 0]])); M[m,2,2]:=subs(sol_sub_seq[m],Matrix([[ 0, 0, 0, 0], [0 , 0, 0, 0], [0 ,0 ,0 ,0 ]])); M[m,2,1]:=subs(sol_sub_seq[m],Matrix([[ 0, 0, 0, 0], [0 , 0, 0, 0], [0 ,0 ,0 ,0 ]])); M[m,1,3]:=subs(sol_sub_seq[m],Matrix([[ 0, 0, 0,f ], [0 ,0 ,0 ,0 ], [0 , 0,e ,0 ]])); M[m,2,3]:=subs(sol_sub_seq[m],Matrix([[0 ,0 ,0 ,g ], [ 0, 0, 0,h ], [0 ,0 ,0 ,0 ]])); M[m,3,1]:=subs(sol_sub_seq[m],Matrix([[ 0, 0, 0,j ], [0 ,0 , 0,k ], [ 0, 0,-1 ,0 ]])); M[m,3,2]:=subs(sol_sub_seq[m],Matrix([[ 0, 0, 0, 0], [0 , 0, 0, 0], [0 ,0 ,0 ,0 ]])); M[m,3,3]:=subs(sol_sub_seq[m],Matrix([[ 0, 0,c ,0 ], [ 0, 0,d ,0 ], [0 ,0 ,0 ,0 ]])); N[m,1,0]:=subs(sol_sub_seq[m],Matrix([[0, p,0 ], [0 ,0 ,1 ], [0 ,0 ,0 ], [0, 0, 0]])); N[m,2,0]:=subs(sol_sub_seq[m],Matrix([[-1, 0, 0], [0, 0,b ], [0 , 0, 0], [0 , 0, 0]])); N[m,3,0]:=subs(sol_sub_seq[m],Matrix([[0, 0, 0], [a ,-1,0 ], [0, 0, 0], [0, 0, 0]])); N[m,1,1]:=subs(sol_sub_seq[m],Matrix([[0, 0, 0], [0, 0, 0], [0 ,0 ,0 ], [0 ,0 ,f ]])); N[m,1,2]:=subs(sol_sub_seq[m],Matrix([[0 , 0,0 ], [0, 0,0 ], [0 ,0 ,0 ], [0 ,0 ,g ]])); N[m,2,1]:=subs(sol_sub_seq[m],Matrix([[0, 0,0 ], [0 ,0 ,0 ], [0 , 0, 0], [ 0, 0,0 ]])); N[m,2,2]:=subs(sol_sub_seq[m],Matrix([[0, 0,0 ], [0 ,0 ,0 ], [0 , 0, 0], [ 0, 0,h ]])); N[m,1,3]:=subs(sol_sub_seq[m],Matrix([[0 ,0 ,0 ], [0 ,0 , 0], [0 , 0, c], [j ,0 ,0 ]])); N[m,2,3]:=subs(sol_sub_seq[m],Matrix([[0 , 0, 0], [0 ,0 ,0 ], [0 ,0 ,d ], [k , 0, 0]])); N[m,3,1]:=subs(sol_sub_seq[m],Matrix([[0, 0,0 ], [0, 0, 0], [0 , 0,e ], [-1 , 0,0 ]])); N[m,3,2]:=subs(sol_sub_seq[m],Matrix([[0, 0,0 ], [0 ,0 ,0 ], [0 , 0, 0], [ 0, 0,0 ]])); N[m,3,3]:=subs(sol_sub_seq[m],Matrix([[0 ,0 ,0 ], [0 ,0 ,0], [-1 ,0 ,0 ], [0 ,0 ,0 ]])); S[m,0]:=Matrix([[0,0,s[m,0,1,3], s[m,0,1,4]], [0,0,s[m,0,2,3], s[m,0,2,4]], [s[m,0,3,1], s[m,0,3,2], 0,0], [s[m,0,4,1], s[m,0,4,2], 0,0]]); for r from 1 by 1 to 3 do S[m,r]:=Matrix([[s[m,r,1,1],s[m,r,1,2],0,0], [s[m,r,2,1],s[m,r,2,2],0,0],[0,0,0,0],[0,0,0,0]]) end do; T[m,3,0]:=add(add(add(x[i,j,k]*(M[m,i,0].S[m,j].N[m,k,0] + M[m,i,j].S[m,0].N[m,k,0] + M[m,i,0].S[m,0].N[m,j,k]), i = 1..2), j = 1..2), k = 1..2); T[m,2,1]:=add(add(x[i,j,3]*(M[m,i,0].S[m,j].N[m,3,0] + M[m,i,j].S[m,0].N[m,3,0] + M[m,i,0].S[m,0].N[m,j,3]), i = 1..2), j = 1..2) +add(add(x[3,j,k]*(M[m,3,0].S[m,j].N[m,k,0] + M[m,3,j].S[m,0].N[m,k,0] + M[m,3,0].S[m,0].N[m,j,k]), j = 1..2), k =1..2) +add(add(x[i,3,k]*(M[m,i,0].S[m,3].N[m,k,0] + M[m,i,3].S[m,0].N[m,k,0] + M[m,i,0].S[m,0].N[m,3,k]), i = 1..2), k =1..2); T[m,1,2]:=add(x[i,3,3]*(M[m,i,0].S[m,3].N[m,3,0] + M[m,i,3].S[m,0].N[m,3,0] + M[m,i,0].S[m,0].N[m,3,3]), i = 1..2) +add(x[3,j,3]*(M[m,3,0].S[m,j].N[m,3,0] + M[m,3,j].S[m,0].N[m,3,0] + M[m,3,0].S[m,0].N[m,j,3]), j = 1..2) +add(x[3,3,k]*(M[m,3,0].S[m,3].N[m,k,0] + M[m,3,3].S[m,0].N[m,k,0] + M[m,3,0].S[m,0].N[m,3,k]), k = 1..2); T[m,0,3]:=x[3,3,3]*(M[m,3,0].S[m,3].N[m,3,0] + M[m,3,3].S[m,0].N[m,3,0] + M[m,3,0].S[m,0].N[m,3,3]); for r from 1 by 1 to 3 do for t from 1 by 1 to 3 do U1[m,3,0][r,t]:= subs(r5d3_sub_list[m], subs(r4d3_sub_list[m], subs(r3d3_sub_list[m], subs(r2d3_sub_list[m],subs(r1d3_sub_list[m], T[m,3,0][r,t]))))) end do end do; for r from 1 by 1 to 3 do for t from 1 by 1 to 3 do U2[m,3,0][r,t]:= subs(r5d3_sub_list[m], subs(r4d3_sub_list[m], subs(r3d3_sub_list[m], subs(r2d3_sub_list[m],subs(r1d3_sub_list[m], U1[m,3,0][r,t]))))) end do end do; for r from 1 by 1 to 3 do for t from 1 by 1 to 3 do U3[m,3,0][r,t]:= subs(r5d3_sub_list[m], subs(r4d3_sub_list[m], subs(r3d3_sub_list[m], subs(r2d3_sub_list[m],subs(r1d3_sub_list[m], U2[m,3,0][r,t]))))) end do end do; for r from 1 by 1 to 3 do for t from 1 by 1 to 3 do U4[m,3,0][r,t]:= subs(r5d3_sub_list[m], subs(r4d3_sub_list[m], subs(r3d3_sub_list[m], subs(r2d3_sub_list[m],subs(r1d3_sub_list[m], U3[m,3,0][r,t]))))) end do end do; for r from 1 by 1 to 3 do for t from 1 by 1 to 3 do U[m,3,0][r,t]:= subs(r5d3_sub_list[m], subs(r4d3_sub_list[m], subs(r3d3_sub_list[m], subs(r2d3_sub_list[m],subs(r1d3_sub_list[m], U4[m,3,0][r,t]))))) end do end do; for r from 1 by 1 to 3 do for t from 1 by 1 to 3 do U1[m,2,1][r,t]:= subs(r5d3_sub_list[m], subs(r4d3_sub_list[m], subs(r3d3_sub_list[m], subs(r2d3_sub_list[m],subs(r1d3_sub_list[m], T[m,2,1][r,t]))))) end do end do; for r from 1 by 1 to 3 do for t from 1 by 1 to 3 do U2[m,2,1][r,t]:= subs(r5d3_sub_list[m], subs(r4d3_sub_list[m], subs(r3d3_sub_list[m], subs(r2d3_sub_list[m],subs(r1d3_sub_list[m], U1[m,2,1][r,t]))))) end do end do; for r from 1 by 1 to 3 do for t from 1 by 1 to 3 do U3[m,2,1][r,t]:= subs(r5d3_sub_list[m], subs(r4d3_sub_list[m], subs(r3d3_sub_list[m], subs(r2d3_sub_list[m],subs(r1d3_sub_list[m], U2[m,2,1][r,t]))))) end do end do; for r from 1 by 1 to 3 do for t from 1 by 1 to 3 do U4[m,2,1][r,t]:= subs(r5d3_sub_list[m], subs(r4d3_sub_list[m], subs(r3d3_sub_list[m], subs(r2d3_sub_list[m],subs(r1d3_sub_list[m], U3[m,2,1][r,t]))))) end do end do; for r from 1 by 1 to 3 do for t from 1 by 1 to 3 do U[m,2,1][r,t]:= subs(r5d3_sub_list[m], subs(r4d3_sub_list[m], subs(r3d3_sub_list[m], subs(r2d3_sub_list[m],subs(r1d3_sub_list[m], U4[m,2,1][r,t]))))) end do end do; for r from 1 by 1 to 3 do for t from 1 by 1 to 3 do U1[m,1,2][r,t]:= subs(r5d3_sub_list[m], subs(r4d3_sub_list[m], subs(r3d3_sub_list[m], subs(r2d3_sub_list[m],subs(r1d3_sub_list[m], T[m,2,1][r,t]))))) end do end do; for r from 1 by 1 to 3 do for t from 1 by 1 to 3 do U2[m,1,2][r,t]:= subs(r5d3_sub_list[m], subs(r4d3_sub_list[m], subs(r3d3_sub_list[m], subs(r2d3_sub_list[m],subs(r1d3_sub_list[m], U1[m,2,1][r,t]))))) end do end do; for r from 1 by 1 to 3 do for t from 1 by 1 to 3 do U3[m,1,2][r,t]:= subs(r5d3_sub_list[m], subs(r4d3_sub_list[m], subs(r3d3_sub_list[m], subs(r2d3_sub_list[m],subs(r1d3_sub_list[m], U2[m,2,1][r,t]))))) end do end do; for r from 1 by 1 to 3 do for t from 1 by 1 to 3 do U4[m,1,2][r,t]:= subs(r5d3_sub_list[m], subs(r4d3_sub_list[m], subs(r3d3_sub_list[m], subs(r2d3_sub_list[m],subs(r1d3_sub_list[m], U3[m,2,1][r,t]))))) end do end do; for r from 1 by 1 to 3 do for t from 1 by 1 to 3 do U[m,1,2][r,t]:= subs(r5d3_sub_list[m], subs(r4d3_sub_list[m], subs(r3d3_sub_list[m], subs(r2d3_sub_list[m],subs(r1d3_sub_list[m], U4[m,2,1][r,t]))))) end do end do; for r from 1 by 1 to 3 do for t from 1 by 1 to 3 do U1[m,0,3][r,t]:= subs(r5d3_sub_list[m], subs(r4d3_sub_list[m], subs(r3d3_sub_list[m], subs(r2d3_sub_list[m],subs(r1d3_sub_list[m], T[m,0,3][r,t]))))) end do end do; for r from 1 by 1 to 3 do for t from 1 by 1 to 3 do U2[m,0,3][r,t]:= subs(r5d3_sub_list[m], subs(r4d3_sub_list[m], subs(r3d3_sub_list[m], subs(r2d3_sub_list[m],subs(r1d3_sub_list[m], U1[m,0,3][r,t]))))) end do end do; for r from 1 by 1 to 3 do for t from 1 by 1 to 3 do U3[m,0,3][r,t]:= subs(r5d3_sub_list[m], subs(r4d3_sub_list[m], subs(r3d3_sub_list[m], subs(r2d3_sub_list[m],subs(r1d3_sub_list[m], U2[m,0,3][r,t]))))) end do end do; for r from 1 by 1 to 3 do for t from 1 by 1 to 3 do U4[m,0,3][r,t]:= subs(r5d3_sub_list[m], subs(r4d3_sub_list[m], subs(r3d3_sub_list[m], subs(r2d3_sub_list[m],subs(r1d3_sub_list[m], U3[m,0,3][r,t]))))) end do end do; for r from 1 by 1 to 3 do for t from 1 by 1 to 3 do U[m,0,3][r,t]:= subs(r5d3_sub_list[m], subs(r4d3_sub_list[m], subs(r3d3_sub_list[m], subs(r2d3_sub_list[m],subs(r1d3_sub_list[m], U4[m,0,3][r,t]))))) end do end do; B[m,3,0]:={seq(seq(seq(seq(seq(coeff(U[m,3,0][r,t], x[u,v,w]), r = 1..3), t = 1..3), u = 1..3), v = 1..3), w=1..3)}; B[m,2,1]:={seq(seq(seq(seq(seq(coeff(U[m,2,1][r,t], x[u,v,w]), r = 1..3), t = 1..3), u = 1..3), v = 1..3), w=1..3)}; B[m,1,2]:={seq(seq(seq(seq(seq(coeff(U[m,1,2][r,t], x[u,v,w]), r = 1..3), t = 1..3), u = 1..3), v = 1..3), w=1..3)}; B[m,0,3]:={seq(seq(seq(seq(seq(coeff(U[m,0,3][r,t], x[u,v,w]), r = 1..3), t = 1..3), u = 1..3), v = 1..3), w=1..3)}; s_solveset[m]:={seq(seq(s[m,0,t,u], t=1..2), u=3..4)} union {seq(seq(s[m,0,t,u], u=1..2), t=3..4)} union {seq(seq(seq(s[m,r,t,u], r=1..3), t=1..2), u=1..2)}; C[m]:=solve(B[m,3,0] union B[m,2,1] union B[m,1,2] union B[m,0,3] union sol_sub_seq[m], s_solveset[m] union {a,b,c,d,e,f,g,h,j,k,p,q}); solution1[m]:=add(x[r]*M[m,r,0].S[m,0], r=1..3) +add(add(x[r,t]*M[m,r,0].S[m,t], r=1..3), t=1..3) +add(add(x[r,t]*M[m,r,t].S[m,0], r=1..3), t=1..3); solution2[m]:=subs(r2d2_sub_list[m],subs(r1d2_sub_list[m], solution1[m])); solution4[m]:=subs(C[m], solution2[m]); solution5[m]:=simplify(solution4[m]) end do; for m from 1 by 1 to nops(sol_sub_seq) do print("______________________________________________"); print("Finding possible d3 for solution", sol_sub_seq[m]); print("______________________________________________"); print(solution5[m]) end do; ################ #Final output from this program is below. ############## "______________________________________________" "Finding possible d3 for solution", {a = -1, b = 1, c = 1, d = 0, e = 0, f = -1 + h, g = 0, h = h, j = 0, k = 0, p = -1, q = -1 } "______________________________________________" [x[3, 3] s[1, 0, 2, 4] , -x[1, 3] s[1, 0, 2, 4] , -x[2] s[1, 0, 2, 4] , x[3] s[1, 0, 2, 4]] [0 , 0 , -x[1] s[1, 0, 2, 4] , x[3] s[1, 0, 2, 4]] [-x[3, 1] s[1, 0, 2, 4] , x[1, 1] s[1, 0, 2, 4] h + x[2, 2] s[1, 0, 2, 4] h - x[1, 1] s[1, 0, 2, 4] , 0 , -x[1] s[1, 0, 2, 4] - x[2] s[1, 0, 2, 4]] "______________________________________________" "Finding possible d3 for solution", {a = -1, b = 1, c = 0, d = 1, 2 e = 0, f = 0, g = RootOf(_Z + 1), h = 0, j = 1, k = 1, p = -1, q = -1} "______________________________________________" [-x[3, 3] s[2, 0, 1, 3] , x[2, 3] s[2, 0, 1, 3] , -x[2] s[2, 0, 1, 3] , -x[3] s[2, 0, 1, 3]] [0 , 0 , -x[1] s[2, 0, 1, 3] , -x[3] s[2, 0, 1, 3]] [[x[1, 3] s[2, 0, 1, 3] + x[2, 3] s[2, 0, 1, 3] - x[3, 1] s[2, 0, 1, 3] , 2 x[1, 2] s[2, 0, 1, 3] RootOf(_Z + 1) , 0 , x[1] s[2, 0, 1, 3] + x[2] s[2, 0, 1, 3]] "______________________________________________" "Finding possible d3 for solution", {a = -%1, b = 1, c = 0, d = -1 - %1, e = -1, f = 0, g = %1, h = 0, j = 0, k = -1 - %1, p = -1 - %1, q = %1} 2 %1 := RootOf(_Z + _Z + 1) "______________________________________________" [x[3, 3] s[3, 0, 3, 1] (%1 + 1) , s[3, 0, 3, 1] (x[3, 1] %1 - x[2, 3]) , %1 x[2] s[3, 0, 3, 1] , -x[3] s[3, 0, 3, 1]] [0 , 0 , x[1] s[3, 0, 3, 1] (%1 + 1) , -x[3] s[3, 0, 3, 1]] [-s[3, 0, 3, 1] (x[3, 1] + x[2, 3] %1 + x[2, 3]) , -x[1, 2] s[3, 0, 3, 1] , 0 , s[3, 0, 3, 1] (x[1] %1 + x[2])] 2 %1 := RootOf(_Z + _Z + 1) "______________________________________________" "Finding possible d3 for solution", {a = -1, b = 1, c = 0, d = 1, 2 e = 0, f = 1 - RootOf(2 _Z - 2 _Z + 1), g = 0, 2 h = RootOf(2 _Z - 2 _Z + 1), j = 0, k = 0, p = -1, q = -1} "______________________________________________" [0 , 0 , -x[2] s[4, 0, 1, 3] , x[3] s[4, 0, 1, 3]] [x[3, 3] s[4, 0, 1, 3] , -s[4, 0, 1, 3] x[2, 3] , -x[1] s[4, 0, 1, 3] , x[3] s[4, 0, 1, 3]] [[-x[3, 1] s[4, 0, 1, 3] , -s[4, 0, 1, 3] ( 2 x[1, 1] RootOf(2 _Z - 2 _Z + 1) 2 - x[2, 2] RootOf(2 _Z - 2 _Z + 1) - x[1, 1]) , 0 , -x[1] s[4, 0, 1, 3] - x[2] s[4, 0, 1, 3]] >