/* This file is part of the paper: "Quadratic Weyl group multiple Dirichlet series of type D_4^(1)" by A. Diaconu, V. Pasol, A.A. Popa It verifies the decompositions in Lemma A.3, of the numerators in the entries of the matrix BB(x,u) = \tilde{B}(1,1,1,1,x;u) The file is written in MAGMA, but it can be easily imported in other software programs. */ R:=FunctionField(Rationals(),2); BB:=Matrix(R, 3,3, [ (-x^32*u^32 - 6*x^32*u^30 - x^32*u^28 - 31*x^30*u^30 - 21*x^30*u^28 - 6*x^28*u^30 + 30*x^30*u^26 - 207*x^28*u^28 - x^30*u^24 + 505*x^28*u^26 - 21*x^26*u^28 - x^30*u^22 - 276*x^28*u^24 - 11*x^26*u^26 - x^24*u^28 + 119*x^28*u^22 + 1457*x^26*u^24 + 505*x^24*u^26 - 31*x^28*u^20 - 2048*x^26*u^22 - 458*x^24*u^24 + 30*x^22*u^26 + 962*x^26*u^20 + 2*x^24*u^22 + 1457*x^22*u^24 - 187*x^26*u^18 - 1743*x^24*u^20 - 6999*x^22*u^22 - 276*x^20*u^24 + 1497*x^24*u^18 + 7291*x^22*u^20 + 2*x^20*u^22 - x^18*u^24 - 330*x^24*u^16 - 2894*x^22*u^18 - 3699*x^20*u^20 - 2048*x^18*u^22 + 990*x^22*u^16 + 11451*x^20*u^18 + 7291*x^18*u^20 + 119*x^16*u^22 - 187*x^22*u^14 - 7640*x^20*u^16 - 4775*x^18*u^18 - 1743*x^16*u^20 - x^14*u^22 + 1497*x^20*u^14 + 1650*x^18*u^16 + 11451*x^16*u^18 + 962*x^14*u^20 - 31*x^20*u^12 - 2894*x^18*u^14 - 21398*x^16*u^16 - 2894*x^14*u^18 - 31*x^12*u^20 + 962*x^18*u^12 + 11451*x^16*u^14 + 1650*x^14*u^16 + 1497*x^12*u^18 - x^18*u^10 - 1743*x^16*u^12 - 4775*x^14*u^14 - 7640*x^12*u^16 - 187*x^10*u^18 + 119*x^16*u^10 + 7291*x^14*u^12 + 11451*x^12*u^14 + 990*x^10*u^16 - 2048*x^14*u^10 - 3699*x^12*u^12 - 2894*x^10*u^14 - 330*x^8*u^16 - x^14*u^8 + 2*x^12*u^10 + 7291*x^10*u^12 + 1497*x^8*u^14 - 276*x^12*u^8 - 6999*x^10*u^10 - 1743*x^8*u^12 - 187*x^6*u^14 + 1457*x^10*u^8 + 2*x^8*u^10 + 962*x^6*u^12 + 30*x^10*u^6 - 458*x^8*u^8 - 2048*x^6*u^10 - 31*x^4*u^12 + 505*x^8*u^6 + 1457*x^6*u^8 + 119*x^4*u^10 - x^8*u^4 - 11*x^6*u^6 - 276*x^4*u^8 - x^2*u^10 - 21*x^6*u^4 + 505*x^4*u^6 - x^2*u^8 - 207*x^4*u^4 + 30*x^2*u^6 - 6*x^4*u^2 - 21*x^2*u^4 - 31*x^2*u^2 - u^4 - 6*u^2 - 1)/(x^38*u^38 - 4*x^38*u^36 + 6*x^38*u^34 - 11*x^36*u^36 - 4*x^38*u^32 + 44*x^36*u^34 - 4*x^34*u^36 + x^38*u^30 - 66*x^36*u^32 + 71*x^34*u^34 + 44*x^36*u^30 - 244*x^34*u^32 + 44*x^32*u^34 - 11*x^36*u^28 + 346*x^34*u^30 - 341*x^32*u^32 + 6*x^30*u^34 - 224*x^34*u^28 + 924*x^32*u^30 - 244*x^30*u^32 + 55*x^34*u^26 - 1166*x^32*u^28 + 1246*x^30*u^30 - 66*x^28*u^32 + 704*x^32*u^26 - 2664*x^30*u^28 + 924*x^28*u^30 - 4*x^26*u^32 - 165*x^32*u^24 + 2866*x^30*u^26 - 3498*x^28*u^28 + 346*x^26*u^30 - 1540*x^30*u^24 + 6072*x^28*u^26 - 2664*x^26*u^28 + 44*x^24*u^30 + 330*x^30*u^22 - 5478*x^28*u^24 + 7738*x^26*u^26 - 1166*x^24*u^28 + x^22*u^30 + 2508*x^28*u^22 - 11092*x^26*u^24 + 6072*x^24*u^26 - 224*x^22*u^28 - 462*x^28*u^20 + 8382*x^26*u^22 - 13838*x^24*u^24 + 2866*x^22*u^26 - 11*x^20*u^28 - 3168*x^26*u^20 + 16412*x^24*u^22 - 11092*x^22*u^24 + 704*x^20*u^26 + 462*x^26*u^18 - 10362*x^24*u^20 + 20318*x^22*u^22 - 5478*x^20*u^24 + 55*x^18*u^26 + 3168*x^24*u^18 - 19888*x^22*u^20 + 16412*x^20*u^22 - 1540*x^18*u^24 - 330*x^24*u^16 + 10362*x^22*u^18 - 24618*x^20*u^20 + 8382*x^18*u^22 - 165*x^16*u^24 - 2508*x^22*u^16 + 19888*x^20*u^18 - 19888*x^18*u^20 + 2508*x^16*u^22 + 165*x^22*u^14 - 8382*x^20*u^16 + 24618*x^18*u^18 - 10362*x^16*u^20 + 330*x^14*u^22 + 1540*x^20*u^14 - 16412*x^18*u^16 + 19888*x^16*u^18 - 3168*x^14*u^20 - 55*x^20*u^12 + 5478*x^18*u^14 - 20318*x^16*u^16 + 10362*x^14*u^18 - 462*x^12*u^20 - 704*x^18*u^12 + 11092*x^16*u^14 - 16412*x^14*u^16 + 3168*x^12*u^18 + 11*x^18*u^10 - 2866*x^16*u^12 + 13838*x^14*u^14 - 8382*x^12*u^16 + 462*x^10*u^18 + 224*x^16*u^10 - 6072*x^14*u^12 + 11092*x^12*u^14 - 2508*x^10*u^16 - x^16*u^8 + 1166*x^14*u^10 - 7738*x^12*u^12 + 5478*x^10*u^14 - 330*x^8*u^16 - 44*x^14*u^8 + 2664*x^12*u^10 - 6072*x^10*u^12 + 1540*x^8*u^14 - 346*x^12*u^8 + 3498*x^10*u^10 - 2866*x^8*u^12 + 165*x^6*u^14 + 4*x^12*u^6 - 924*x^10*u^8 + 2664*x^8*u^10 - 704*x^6*u^12 + 66*x^10*u^6 - 1246*x^8*u^8 + 1166*x^6*u^10 - 55*x^4*u^12 + 244*x^8*u^6 - 924*x^6*u^8 + 224*x^4*u^10 - 6*x^8*u^4 + 341*x^6*u^6 - 346*x^4*u^8 + 11*x^2*u^10 - 44*x^6*u^4 + 244*x^4*u^6 - 44*x^2*u^8 - 71*x^4*u^4 + 66*x^2*u^6 - u^8 + 4*x^4*u^2 - 44*x^2*u^4 + 4*u^6 + 11*x^2*u^2 - 6*u^4 + 4*u^2 - 1), (-x^15*u^23 - 6*x^15*u^21 - x^15*u^19 - 31*x^13*u^21 - 21*x^13*u^19 + 30*x^13*u^17 - 187*x^11*u^19 - x^13*u^15 + 495*x^11*u^17 - x^13*u^13 - 276*x^11*u^15 - 330*x^9*u^17 + 119*x^11*u^13 + 1518*x^9*u^15 - 31*x^11*u^11 - 2057*x^9*u^13 - 187*x^7*u^15 + 968*x^9*u^11 + 968*x^7*u^13 - 187*x^9*u^9 - 2057*x^7*u^11 - 31*x^5*u^13 + 1518*x^7*u^9 + 119*x^5*u^11 - 330*x^7*u^7 - 276*x^5*u^9 - x^3*u^11 + 495*x^5*u^7 - x^3*u^9 - 187*x^5*u^5 + 30*x^3*u^7 - 21*x^3*u^5 - 31*x^3*u^3 - x*u^5 - 6*x*u^3 - x*u)/(x^22*u^30 - 4*x^22*u^28 + 6*x^22*u^26 - 11*x^20*u^28 - 4*x^22*u^24 + 44*x^20*u^26 + x^22*u^22 - 66*x^20*u^24 + 55*x^18*u^26 + 44*x^20*u^22 - 220*x^18*u^24 - 11*x^20*u^20 + 330*x^18*u^22 - 165*x^16*u^24 - 220*x^18*u^20 + 660*x^16*u^22 + 55*x^18*u^18 - 990*x^16*u^20 + 330*x^14*u^22 + 660*x^16*u^18 - 1320*x^14*u^20 - 165*x^16*u^16 + 1980*x^14*u^18 - 462*x^12*u^20 - 1320*x^14*u^16 + 1848*x^12*u^18 + 330*x^14*u^14 - 2772*x^12*u^16 + 462*x^10*u^18 + 1848*x^12*u^14 - 1848*x^10*u^16 - 462*x^12*u^12 + 2772*x^10*u^14 - 330*x^8*u^16 - 1848*x^10*u^12 + 1320*x^8*u^14 + 462*x^10*u^10 - 1980*x^8*u^12 + 165*x^6*u^14 + 1320*x^8*u^10 - 660*x^6*u^12 - 330*x^8*u^8 + 990*x^6*u^10 - 55*x^4*u^12 - 660*x^6*u^8 + 220*x^4*u^10 + 165*x^6*u^6 - 330*x^4*u^8 + 11*x^2*u^10 + 220*x^4*u^6 - 44*x^2*u^8 - 55*x^4*u^4 + 66*x^2*u^6 - u^8 - 44*x^2*u^4 + 4*u^6 + 11*x^2*u^2 - 6*u^4 + 4*u^2 - 1), (-4*x^28*u^29 - 20*x^28*u^27 - 80*x^26*u^27 + 40*x^26*u^25 - 4*x^24*u^27 + 60*x^26*u^23 - 300*x^24*u^25 + 1140*x^24*u^23 + 60*x^22*u^25 - 4*x^26*u^19 - 960*x^24*u^21 + 372*x^24*u^19 + 1340*x^22*u^21 + 704*x^20*u^23 - 80*x^24*u^17 - 2792*x^22*u^19 - 1796*x^20*u^21 - 16*x^18*u^23 + 1556*x^22*u^17 + 788*x^20*u^19 + 1004*x^18*u^21 - 300*x^22*u^15 - 1024*x^20*u^17 - 5932*x^18*u^19 - 392*x^16*u^21 + 1180*x^20*u^15 + 8532*x^18*u^17 + 792*x^16*u^19 + 4*x^14*u^21 - 300*x^20*u^13 - 3840*x^18*u^15 - 1428*x^16*u^17 - 1120*x^14*u^19 + 740*x^18*u^13 + 5520*x^16*u^15 + 5016*x^14*u^17 + 80*x^12*u^19 - 80*x^18*u^11 - 5016*x^16*u^13 - 5520*x^14*u^15 - 740*x^12*u^17 + 1120*x^16*u^11 + 1428*x^14*u^13 + 3840*x^12*u^15 + 300*x^10*u^17 - 4*x^16*u^9 - 792*x^14*u^11 - 8532*x^12*u^13 - 1180*x^10*u^15 + 392*x^14*u^9 + 5932*x^12*u^11 + 1024*x^10*u^13 + 300*x^8*u^15 - 1004*x^12*u^9 - 788*x^10*u^11 - 1556*x^8*u^13 + 16*x^12*u^7 + 1796*x^10*u^9 + 2792*x^8*u^11 + 80*x^6*u^13 - 704*x^10*u^7 - 1340*x^8*u^9 - 372*x^6*u^11 + 960*x^6*u^9 + 4*x^4*u^11 - 60*x^8*u^5 - 1140*x^6*u^7 + 300*x^6*u^5 - 60*x^4*u^7 + 4*x^6*u^3 - 40*x^4*u^5 + 80*x^4*u^3 + 20*x^2*u^3 + 4*x^2*u)/(x^36*u^36 - 4*x^36*u^34 + 6*x^36*u^32 - 10*x^34*u^34 - 4*x^36*u^30 + 40*x^34*u^32 - 4*x^32*u^34 + x^36*u^28 - 60*x^34*u^30 + 61*x^32*u^32 + 40*x^34*u^28 - 204*x^32*u^30 + 40*x^30*u^32 - 10*x^34*u^26 + 286*x^32*u^28 - 280*x^30*u^30 + 6*x^28*u^32 - 184*x^32*u^26 + 720*x^30*u^28 - 204*x^28*u^30 + 45*x^32*u^24 - 880*x^30*u^26 + 966*x^28*u^28 - 60*x^26*u^30 + 520*x^30*u^24 - 1944*x^28*u^26 + 720*x^26*u^28 - 4*x^24*u^30 - 120*x^30*u^22 + 1986*x^28*u^24 - 2532*x^26*u^26 + 286*x^24*u^28 - 1020*x^28*u^22 + 4128*x^26*u^24 - 1944*x^24*u^26 + 40*x^22*u^28 + 210*x^28*u^20 - 3492*x^26*u^22 + 5206*x^24*u^24 - 880*x^22*u^26 + x^20*u^28 + 1488*x^26*u^20 - 6964*x^24*u^22 + 4128*x^22*u^24 - 184*x^20*u^26 - 252*x^26*u^18 + 4890*x^24*u^20 - 8632*x^22*u^22 + 1986*x^20*u^24 - 10*x^18*u^26 - 1680*x^24*u^18 + 9448*x^22*u^20 - 6964*x^20*u^22 + 520*x^18*u^24 + 210*x^24*u^16 - 5472*x^22*u^18 + 11686*x^20*u^20 - 3492*x^18*u^22 + 45*x^16*u^24 + 1488*x^22*u^16 - 10440*x^20*u^18 + 9448*x^18*u^20 - 1020*x^16*u^22 - 120*x^22*u^14 + 4890*x^20*u^16 - 12932*x^18*u^18 + 4890*x^16*u^20 - 120*x^14*u^22 - 1020*x^20*u^14 + 9448*x^18*u^16 - 10440*x^16*u^18 + 1488*x^14*u^20 + 45*x^20*u^12 - 3492*x^18*u^14 + 11686*x^16*u^16 - 5472*x^14*u^18 + 210*x^12*u^20 + 520*x^18*u^12 - 6964*x^16*u^14 + 9448*x^14*u^16 - 1680*x^12*u^18 - 10*x^18*u^10 + 1986*x^16*u^12 - 8632*x^14*u^14 + 4890*x^12*u^16 - 252*x^10*u^18 - 184*x^16*u^10 + 4128*x^14*u^12 - 6964*x^12*u^14 + 1488*x^10*u^16 + x^16*u^8 - 880*x^14*u^10 + 5206*x^12*u^12 - 3492*x^10*u^14 + 210*x^8*u^16 + 40*x^14*u^8 - 1944*x^12*u^10 + 4128*x^10*u^12 - 1020*x^8*u^14 + 286*x^12*u^8 - 2532*x^10*u^10 + 1986*x^8*u^12 - 120*x^6*u^14 - 4*x^12*u^6 + 720*x^10*u^8 - 1944*x^8*u^10 + 520*x^6*u^12 - 60*x^10*u^6 + 966*x^8*u^8 - 880*x^6*u^10 + 45*x^4*u^12 - 204*x^8*u^6 + 720*x^6*u^8 - 184*x^4*u^10 + 6*x^8*u^4 - 280*x^6*u^6 + 286*x^4*u^8 - 10*x^2*u^10 + 40*x^6*u^4 - 204*x^4*u^6 + 40*x^2*u^8 + 61*x^4*u^4 - 60*x^2*u^6 + u^8 - 4*x^4*u^2 + 40*x^2*u^4 - 4*u^6 - 10*x^2*u^2 + 6*u^4 - 4*u^2 + 1), (-x^31*u^23 - 31*x^29*u^21 - 6*x^27*u^21 - 187*x^27*u^19 - 21*x^25*u^19 - 330*x^25*u^17 - x^23*u^19 + 495*x^23*u^17 - 187*x^23*u^15 + 30*x^21*u^17 + 1518*x^21*u^15 - 31*x^21*u^13 - 276*x^19*u^15 + 968*x^19*u^13 - x^17*u^15 - x^19*u^11 - 2057*x^17*u^13 + 119*x^17*u^11 + 119*x^15*u^13 - 2057*x^15*u^11 - x^13*u^13 - x^15*u^9 + 968*x^13*u^11 - 276*x^13*u^9 - 31*x^11*u^11 + 1518*x^11*u^9 + 30*x^11*u^7 - 187*x^9*u^9 + 495*x^9*u^7 - x^9*u^5 - 330*x^7*u^7 - 21*x^7*u^5 - 187*x^5*u^5 - 6*x^5*u^3 - 31*x^3*u^3 - x*u)/(x^38*u^30 - 11*x^36*u^28 - 4*x^34*u^28 + 55*x^34*u^26 + 44*x^32*u^26 - 165*x^32*u^24 + 6*x^30*u^26 - 220*x^30*u^24 + 330*x^30*u^22 - 66*x^28*u^24 + 660*x^28*u^22 - 4*x^26*u^24 - 462*x^28*u^20 + 330*x^26*u^22 - 1320*x^26*u^20 + 44*x^24*u^22 + 462*x^26*u^18 - 990*x^24*u^20 + x^22*u^22 + 1848*x^24*u^18 - 220*x^22*u^20 - 330*x^24*u^16 + 1980*x^22*u^18 - 11*x^20*u^20 - 1848*x^22*u^16 + 660*x^20*u^18 + 165*x^22*u^14 - 2772*x^20*u^16 + 55*x^18*u^18 + 1320*x^20*u^14 - 1320*x^18*u^16 - 55*x^20*u^12 + 2772*x^18*u^14 - 165*x^16*u^16 - 660*x^18*u^12 + 1848*x^16*u^14 + 11*x^18*u^10 - 1980*x^16*u^12 + 330*x^14*u^14 + 220*x^16*u^10 - 1848*x^14*u^12 - x^16*u^8 + 990*x^14*u^10 - 462*x^12*u^12 - 44*x^14*u^8 + 1320*x^12*u^10 - 330*x^12*u^8 + 462*x^10*u^10 + 4*x^12*u^6 - 660*x^10*u^8 + 66*x^10*u^6 - 330*x^8*u^8 + 220*x^8*u^6 - 6*x^8*u^4 + 165*x^6*u^6 - 44*x^6*u^4 - 55*x^4*u^4 + 4*x^4*u^2 + 11*x^2*u^2 - 1), (-x^14*u^14 - 31*x^12*u^12 - 187*x^10*u^10 - 330*x^8*u^8 - 187*x^6*u^6 - 31*x^4*u^4 - x^2*u^2)/(x^22*u^22 - 11*x^20*u^20 + 55*x^18*u^18 - 165*x^16*u^16 + 330*x^14*u^14 - 462*x^12*u^12 + 462*x^10*u^10 - 330*x^8*u^8 + 165*x^6*u^6 - 55*x^4*u^4 + 11*x^2*u^2 - 1), (-4*x^27*u^20 - 80*x^25*u^18 - 4*x^23*u^18 - 300*x^23*u^16 + 60*x^21*u^16 - 300*x^21*u^14 + 700*x^19*u^14 - 80*x^19*u^12 - 16*x^17*u^14 + 1140*x^17*u^12 - 4*x^17*u^10 - 392*x^15*u^12 + 392*x^15*u^10 + 4*x^13*u^12 - 1140*x^13*u^10 + 16*x^13*u^8 + 80*x^11*u^10 - 700*x^11*u^8 + 300*x^9*u^8 - 60*x^9*u^6 + 300*x^7*u^6 + 4*x^7*u^4 + 80*x^5*u^4 + 4*x^3*u^2)/(x^36*u^28 - 10*x^34*u^26 - 4*x^32*u^26 + 45*x^32*u^24 + 40*x^30*u^24 - 120*x^30*u^22 + 6*x^28*u^24 - 180*x^28*u^22 + 210*x^28*u^20 - 60*x^26*u^22 + 480*x^26*u^20 - 4*x^24*u^22 - 252*x^26*u^18 + 270*x^24*u^20 - 840*x^24*u^18 + 40*x^22*u^20 + 210*x^24*u^16 - 720*x^22*u^18 + x^20*u^20 + 1008*x^22*u^16 - 180*x^20*u^18 - 120*x^22*u^14 + 1260*x^20*u^16 - 10*x^18*u^18 - 840*x^20*u^14 + 480*x^18*u^16 + 45*x^20*u^12 - 1512*x^18*u^14 + 45*x^16*u^16 + 480*x^18*u^12 - 840*x^16*u^14 - 10*x^18*u^10 + 1260*x^16*u^12 - 120*x^14*u^14 - 180*x^16*u^10 + 1008*x^14*u^12 + x^16*u^8 - 720*x^14*u^10 + 210*x^12*u^12 + 40*x^14*u^8 - 840*x^12*u^10 + 270*x^12*u^8 - 252*x^10*u^10 - 4*x^12*u^6 + 480*x^10*u^8 - 60*x^10*u^6 + 210*x^8*u^8 - 180*x^8*u^6 + 6*x^8*u^4 - 120*x^6*u^6 + 40*x^6*u^4 + 45*x^4*u^4 - 4*x^4*u^2 - 10*x^2*u^2 + 1), (-4*x^30*u^29 - 4*x^30*u^27 - 80*x^28*u^27 + 60*x^28*u^25 - 20*x^26*u^27 - 16*x^28*u^23 - 300*x^26*u^25 + 4*x^28*u^21 + 704*x^26*u^23 + 40*x^24*u^25 - 392*x^26*u^21 + 80*x^26*u^19 + 1004*x^24*u^21 + 1140*x^22*u^23 - 1120*x^24*u^19 - 1796*x^22*u^21 + 60*x^20*u^23 + 300*x^24*u^17 + 792*x^22*u^19 + 1340*x^20*u^21 - 740*x^22*u^17 - 5932*x^20*u^19 - 960*x^18*u^21 + 300*x^22*u^15 + 5016*x^20*u^17 + 788*x^18*u^19 - 1180*x^20*u^15 - 1428*x^18*u^17 - 2792*x^16*u^19 + 80*x^20*u^13 + 3840*x^18*u^15 + 8532*x^16*u^17 + 372*x^14*u^19 - 1556*x^18*u^13 - 5520*x^16*u^15 - 1024*x^14*u^17 - 4*x^12*u^19 + 4*x^18*u^11 + 1024*x^16*u^13 + 5520*x^14*u^15 + 1556*x^12*u^17 - 372*x^16*u^11 - 8532*x^14*u^13 - 3840*x^12*u^15 - 80*x^10*u^17 + 2792*x^14*u^11 + 1428*x^12*u^13 + 1180*x^10*u^15 - 788*x^12*u^11 - 5016*x^10*u^13 - 300*x^8*u^15 + 960*x^12*u^9 + 5932*x^10*u^11 + 740*x^8*u^13 - 1340*x^10*u^9 - 792*x^8*u^11 - 300*x^6*u^13 - 60*x^10*u^7 + 1796*x^8*u^9 + 1120*x^6*u^11 - 1140*x^8*u^7 - 1004*x^6*u^9 - 80*x^4*u^11 + 392*x^4*u^9 - 40*x^6*u^5 - 704*x^4*u^7 - 4*x^2*u^9 + 300*x^4*u^5 + 16*x^2*u^7 + 20*x^4*u^3 - 60*x^2*u^5 + 80*x^2*u^3 + 4*u^3 + 4*u)/(x^36*u^36 - 4*x^36*u^34 + 6*x^36*u^32 - 10*x^34*u^34 - 4*x^36*u^30 + 40*x^34*u^32 - 4*x^32*u^34 + x^36*u^28 - 60*x^34*u^30 + 61*x^32*u^32 + 40*x^34*u^28 - 204*x^32*u^30 + 40*x^30*u^32 - 10*x^34*u^26 + 286*x^32*u^28 - 280*x^30*u^30 + 6*x^28*u^32 - 184*x^32*u^26 + 720*x^30*u^28 - 204*x^28*u^30 + 45*x^32*u^24 - 880*x^30*u^26 + 966*x^28*u^28 - 60*x^26*u^30 + 520*x^30*u^24 - 1944*x^28*u^26 + 720*x^26*u^28 - 4*x^24*u^30 - 120*x^30*u^22 + 1986*x^28*u^24 - 2532*x^26*u^26 + 286*x^24*u^28 - 1020*x^28*u^22 + 4128*x^26*u^24 - 1944*x^24*u^26 + 40*x^22*u^28 + 210*x^28*u^20 - 3492*x^26*u^22 + 5206*x^24*u^24 - 880*x^22*u^26 + x^20*u^28 + 1488*x^26*u^20 - 6964*x^24*u^22 + 4128*x^22*u^24 - 184*x^20*u^26 - 252*x^26*u^18 + 4890*x^24*u^20 - 8632*x^22*u^22 + 1986*x^20*u^24 - 10*x^18*u^26 - 1680*x^24*u^18 + 9448*x^22*u^20 - 6964*x^20*u^22 + 520*x^18*u^24 + 210*x^24*u^16 - 5472*x^22*u^18 + 11686*x^20*u^20 - 3492*x^18*u^22 + 45*x^16*u^24 + 1488*x^22*u^16 - 10440*x^20*u^18 + 9448*x^18*u^20 - 1020*x^16*u^22 - 120*x^22*u^14 + 4890*x^20*u^16 - 12932*x^18*u^18 + 4890*x^16*u^20 - 120*x^14*u^22 - 1020*x^20*u^14 + 9448*x^18*u^16 - 10440*x^16*u^18 + 1488*x^14*u^20 + 45*x^20*u^12 - 3492*x^18*u^14 + 11686*x^16*u^16 - 5472*x^14*u^18 + 210*x^12*u^20 + 520*x^18*u^12 - 6964*x^16*u^14 + 9448*x^14*u^16 - 1680*x^12*u^18 - 10*x^18*u^10 + 1986*x^16*u^12 - 8632*x^14*u^14 + 4890*x^12*u^16 - 252*x^10*u^18 - 184*x^16*u^10 + 4128*x^14*u^12 - 6964*x^12*u^14 + 1488*x^10*u^16 + x^16*u^8 - 880*x^14*u^10 + 5206*x^12*u^12 - 3492*x^10*u^14 + 210*x^8*u^16 + 40*x^14*u^8 - 1944*x^12*u^10 + 4128*x^10*u^12 - 1020*x^8*u^14 + 286*x^12*u^8 - 2532*x^10*u^10 + 1986*x^8*u^12 - 120*x^6*u^14 - 4*x^12*u^6 + 720*x^10*u^8 - 1944*x^8*u^10 + 520*x^6*u^12 - 60*x^10*u^6 + 966*x^8*u^8 - 880*x^6*u^10 + 45*x^4*u^12 - 204*x^8*u^6 + 720*x^6*u^8 - 184*x^4*u^10 + 6*x^8*u^4 - 280*x^6*u^6 + 286*x^4*u^8 - 10*x^2*u^10 + 40*x^6*u^4 - 204*x^4*u^6 + 40*x^2*u^8 + 61*x^4*u^4 - 60*x^2*u^6 + u^8 - 4*x^4*u^2 + 40*x^2*u^4 - 4*u^6 - 10*x^2*u^2 + 6*u^4 - 4*u^2 + 1), (-4*x^13*u^20 - 4*x^13*u^18 - 80*x^11*u^18 + 60*x^11*u^16 - 16*x^11*u^14 - 300*x^9*u^16 + 4*x^11*u^12 + 700*x^9*u^14 - 392*x^9*u^12 - 300*x^7*u^14 + 80*x^9*u^10 + 1140*x^7*u^12 - 1140*x^7*u^10 - 80*x^5*u^12 + 300*x^7*u^8 + 392*x^5*u^10 - 700*x^5*u^8 - 4*x^3*u^10 + 300*x^5*u^6 + 16*x^3*u^8 - 60*x^3*u^6 + 80*x^3*u^4 + 4*x*u^4 + 4*x*u^2)/(x^20*u^28 - 4*x^20*u^26 + 6*x^20*u^24 - 10*x^18*u^26 - 4*x^20*u^22 + 40*x^18*u^24 + x^20*u^20 - 60*x^18*u^22 + 45*x^16*u^24 + 40*x^18*u^20 - 180*x^16*u^22 - 10*x^18*u^18 + 270*x^16*u^20 - 120*x^14*u^22 - 180*x^16*u^18 + 480*x^14*u^20 + 45*x^16*u^16 - 720*x^14*u^18 + 210*x^12*u^20 + 480*x^14*u^16 - 840*x^12*u^18 - 120*x^14*u^14 + 1260*x^12*u^16 - 252*x^10*u^18 - 840*x^12*u^14 + 1008*x^10*u^16 + 210*x^12*u^12 - 1512*x^10*u^14 + 210*x^8*u^16 + 1008*x^10*u^12 - 840*x^8*u^14 - 252*x^10*u^10 + 1260*x^8*u^12 - 120*x^6*u^14 - 840*x^8*u^10 + 480*x^6*u^12 + 210*x^8*u^8 - 720*x^6*u^10 + 45*x^4*u^12 + 480*x^6*u^8 - 180*x^4*u^10 - 120*x^6*u^6 + 270*x^4*u^8 - 10*x^2*u^10 - 180*x^4*u^6 + 40*x^2*u^8 + 45*x^4*u^4 - 60*x^2*u^6 + u^8 + 40*x^2*u^4 - 4*u^6 - 10*x^2*u^2 + 6*u^4 - 4*u^2 + 1), (-15*x^26*u^26 - 10*x^26*u^24 + x^26*u^22 - 180*x^24*u^24 + 231*x^24*u^22 - 10*x^22*u^24 - 74*x^24*u^20 - 358*x^22*u^22 + 15*x^24*u^18 + 1182*x^22*u^20 + 231*x^20*u^22 - 834*x^22*u^18 - 144*x^20*u^20 + x^18*u^22 + 180*x^22*u^16 + 725*x^20*u^18 + 1182*x^18*u^20 - 1166*x^20*u^16 - 2789*x^18*u^18 - 74*x^16*u^20 + 378*x^20*u^14 + 1464*x^18*u^16 + 725*x^16*u^18 - 462*x^18*u^14 - 3852*x^16*u^16 - 834*x^14*u^18 + 180*x^18*u^12 + 4336*x^16*u^14 + 1464*x^14*u^16 + 15*x^12*u^18 - 1166*x^16*u^12 - 684*x^14*u^14 - 1166*x^12*u^16 + 15*x^16*u^10 + 1464*x^14*u^12 + 4336*x^12*u^14 + 180*x^10*u^16 - 834*x^14*u^10 - 3852*x^12*u^12 - 462*x^10*u^14 + 725*x^12*u^10 + 1464*x^10*u^12 + 378*x^8*u^14 - 74*x^12*u^8 - 2789*x^10*u^10 - 1166*x^8*u^12 + 1182*x^10*u^8 + 725*x^8*u^10 + 180*x^6*u^12 + x^10*u^6 - 144*x^8*u^8 - 834*x^6*u^10 + 231*x^8*u^6 + 1182*x^6*u^8 + 15*x^4*u^10 - 358*x^6*u^6 - 74*x^4*u^8 - 10*x^6*u^4 + 231*x^4*u^6 - 180*x^4*u^4 + x^2*u^6 - 10*x^2*u^4 - 15*x^2*u^2)/(x^34*u^34 - 4*x^34*u^32 + 6*x^34*u^30 - 9*x^32*u^32 - 4*x^34*u^28 + 36*x^32*u^30 - 4*x^30*u^32 + x^34*u^26 - 54*x^32*u^28 + 52*x^30*u^30 + 36*x^32*u^26 - 168*x^30*u^28 + 36*x^28*u^30 - 9*x^32*u^24 + 232*x^30*u^26 - 228*x^28*u^28 + 6*x^26*u^30 - 148*x^30*u^24 + 552*x^28*u^26 - 168*x^26*u^28 + 36*x^30*u^22 - 648*x^28*u^24 + 738*x^26*u^26 - 54*x^24*u^28 + 372*x^28*u^22 - 1392*x^26*u^24 + 552*x^24*u^26 - 4*x^22*u^28 - 84*x^28*u^20 + 1338*x^26*u^22 - 1794*x^24*u^24 + 232*x^22*u^26 - 648*x^26*u^20 + 2736*x^24*u^22 - 1392*x^22*u^24 + 36*x^20*u^26 + 126*x^26*u^18 - 2154*x^24*u^20 + 3412*x^22*u^22 - 648*x^20*u^24 + x^18*u^26 + 840*x^24*u^18 - 4228*x^22*u^20 + 2736*x^20*u^22 - 148*x^18*u^24 - 126*x^24*u^16 + 2736*x^22*u^18 - 5220*x^20*u^20 + 1338*x^18*u^22 - 9*x^16*u^24 - 840*x^22*u^16 + 5220*x^20*u^18 - 4228*x^18*u^20 + 372*x^16*u^22 + 84*x^22*u^14 - 2736*x^20*u^16 + 6466*x^18*u^18 - 2154*x^16*u^20 + 36*x^14*u^22 + 648*x^20*u^14 - 5220*x^18*u^16 + 5220*x^16*u^18 - 648*x^14*u^20 - 36*x^20*u^12 + 2154*x^18*u^14 - 6466*x^16*u^16 + 2736*x^14*u^18 - 84*x^12*u^20 - 372*x^18*u^12 + 4228*x^16*u^14 - 5220*x^14*u^16 + 840*x^12*u^18 + 9*x^18*u^10 - 1338*x^16*u^12 + 5220*x^14*u^14 - 2736*x^12*u^16 + 126*x^10*u^18 + 148*x^16*u^10 - 2736*x^14*u^12 + 4228*x^12*u^14 - 840*x^10*u^16 - x^16*u^8 + 648*x^14*u^10 - 3412*x^12*u^12 + 2154*x^10*u^14 - 126*x^8*u^16 - 36*x^14*u^8 + 1392*x^12*u^10 - 2736*x^10*u^12 + 648*x^8*u^14 - 232*x^12*u^8 + 1794*x^10*u^10 - 1338*x^8*u^12 + 84*x^6*u^14 + 4*x^12*u^6 - 552*x^10*u^8 + 1392*x^8*u^10 - 372*x^6*u^12 + 54*x^10*u^6 - 738*x^8*u^8 + 648*x^6*u^10 - 36*x^4*u^12 + 168*x^8*u^6 - 552*x^6*u^8 + 148*x^4*u^10 - 6*x^8*u^4 + 228*x^6*u^6 - 232*x^4*u^8 + 9*x^2*u^10 - 36*x^6*u^4 + 168*x^4*u^6 - 36*x^2*u^8 - 52*x^4*u^4 + 54*x^2*u^6 - u^8 + 4*x^4*u^2 - 36*x^2*u^4 + 4*u^6 + 9*x^2*u^2 - 6*u^4 + 4*u^2 - 1) ]); p:=x^2; q:=u^2; r:=p*q; //the sequences ak, ek, fk below, are the vectors \vec{a}_k, \vec{e}_k, \vec{f}_k //in Lemma A.3. a4:=[(r-1)^(4-2*a)*(q-1)^a*(p*r-1)^a: a in [0..2]]; a6:=[(r-1)^(6-2*a)*(q-1)^a*(p*r-1)^a: a in [0..3]]; a8:=[(r-1)^(8-2*a)*(q-1)^a*(p*r-1)^a: a in [0..4]]; e3:=[(r-1)^(3-a)*(q-1)^a: a in [0..3]]; e4:=[(r-1)^(4-a)*(q-1)^a: a in [0..4]]; f3:=[(r-1)^(3-i)*(p*r-1)^i: i in [0..3]]; f4:=[(r-1)^(4-i)*(p*r-1)^i: i in [0..4]]; b11:=BB[1,1]*(q-1)^4*(r-1)^11*(p*r-1)^4; p11:=[-8*p^8*q^7 - 34*r^7 - 54*p^7*q^6 - 8*p^6*q^7 - 368*r^6 - 54*p^5*q^6 - 786*r^5 - 1088*r^4 - 786*r^3 - 54*p^3*q^2 - 368*r^2 - 8*p^2*q - 54*p*q^2 - 34*r - 8*q, -22*p^7*q^6 - 70*r^6 - 38*p^6*q^5 - 22*p^5*q^6 - 912*r^5 - 38*p^4*q^5 - 1252*r^4 - 38*p^4*q^3 - 912*r^3 - 22*p^3*q^2 - 38*p^2*q^3 - 70*r^2 - 22*p*q^2, -r^8 - 5*p^7*q^6 - 25*p^6*q^5 - 5*p^5*q^6 - 466*r^5 - 25*p^4*q^5 - 994*r^4 - 25*p^4*q^3 - 466*r^3 - 5*p^3*q^2 - 25*p^2*q^3 - 5*p*q^2 - 1, -140*r^5 - 488*r^4 - 140*r^3, -r^7 - 31*r^6 - 187*r^5 - 330*r^4 - 187*r^3 - 31*r^2 - r]; b11 eq &+[ a8[i]*p11[i] : i in [1..5] ];//x eq y return true if x=y and false otherwise //&+s returns the sum of the sequence s b12:=BB[1,2]/u/x*(q-1)^4*(r-1)^11; p12:=[-8*r^3 - 56*r^2 - 24*p*q^2 - 32*r - 8*q, -24*r^4 - 200*r^3 - 200*r^2 - 24*p*q^2, -25*r^5 - 269*r^4 - 426*r^3 - 15*p^2*q^3 - 27*r^2 - 5*p*q^2 - 1, -10*r^6 - 155*r^5 - 408*r^4 - 195*r^3, -r^7 - 31*r^6 - 187*r^5 - 330*r^4 - 187*r^3 - 31*r^2 - r]; b12 eq &+[ e4[i]*p12[i] : i in [1..5] ]; b21:=BB[2,1]/u/x*(r-1)^11*(p*r-1)^4; p21:=[-32*p^10*q^6 - 8*p^9*q^6 - 56*p^9*q^5 - 24*p^8*q^5 - 8*p^8*q^4, -200*p^8*q^5 - 24*p^7*q^5 - 200*p^7*q^4 - 24*p^6*q^3, -p^9*q^7 - 27*p^7*q^5 - 5*p^6*q^5 - 426*p^6*q^4 - 15*p^5*q^4 - 269*p^5*q^3 - 25*p^4*q^2, -195*p^5*q^4 - 408*p^4*q^3 - 155*p^3*q^2 - 10*p^2*q, -p^6*q^6 - 31*p^5*q^5 - 187*p^4*q^4 - 330*p^3*q^3 - 187*p^2*q^2 - 31*p*q - 1]; b21 eq &+[f4[i]*p21[i] : i in [1..5]]; b32:=BB[3,2]/x*(q-1)^4*(r-1)^10; p32:= [-2*r^3 - 14*r^2 - 14*r - q - 1, -5*r^4 - 44*r^3 - 56*r^2 - 7*r, -4*r^5 - 49*r^4 - 99*r^3 - 39*r^2 - r, -r^6 - 20*r^5 - 75*r^4 - 75*r^3 - 20*r^2 - r]; b32 eq 4*q*(&+[ e3[i]*p32[i] : i in [1..4] ]); b23:=BB[2,3]/x*(r-1)^10*(p*r-1)^4; p23:=[-p^9*q^6 - 14*p^8*q^5 - p^7*q^5 - 14*p^7*q^4 - 2*p^6*q^3, -7*p^7*q^5 - 56*p^6*q^4 - 44*p^5*q^3 - 5*p^4*q^2, -p^6*q^5 - 39*p^5*q^4 - 99*p^4*q^3 - 49*p^3*q^2 - 4*p^2*q, -p^5*q^5 - 20*p^4*q^4 - 75*p^3*q^3 - 75*p^2*q^2 - 20*p*q - 1]; b23 eq 4*p*q*&+[f3[i]*p23[i] : i in [1..4]]; b33:=BB[3,3]*(q-1)^4*(r-1)^9*(p*r-1)^4; p33:=[-15*r^6 - 9*p^6*q^5 - 152*r^5 - 9*p^4*q^5 - 383*r^4 - 528*r^3 - 383*r^2 - 9*p^2*q - 152*r - 9*q - 15, -p^6*q^5 - 78*r^5 - p^4*q^5 - 496*r^4 - 768*r^3 - 496*r^2 - p^2*q - 78*r - q, -25*r^5 - 350*r^4 - 658*r^3 - 350*r^2 - 25*r, -15*r^5 - 180*r^4 - 378*r^3 - 180*r^2 - 15*r ]; b33 eq p*q*(&+[ a6[i]*p33[i] : i in [1..4] ]); b13:=BB[1,3]/u*(q-1)^4*(r-1)^10*(p*r-1)^4; b31:=BB[3,1]/u*(q-1)^4*(r-1)^10*(p*r-1)^4; p13:=[-r^6 - p^6*q^5 - 12*r^5 - p^4*q^5 - 29*r^4 - 40*r^3 - 29*r^2 - p^2*q - 12*r - q - 1, -6*r^5 - 42*r^4 - 64*r^3 - 42*r^2 - 6*r, -r^5 - 34*r^4 - 66*r^3 - 34*r^2 - r, -r^5 - 19*r^4 - 56*r^3 - 19*r^2 - r ]; q13:=[ -5*r^4 - p^4*q^3 - 26*r^3 - p^2*q^3 - 38*r^2 - p^2*q - 26*r - q - 5, -19*r^3 - 50*r^2 - 19*r, -5*r^3 - 30*r^2 - 5*r]; b13 eq 4*(q-1)*p*(r+1)*(&+[ a6[i]*p13[i] : i in [1..4] ])+ 4*(r-1)^3*(p+1)*r*(r+1)*(&+[ a4[i]*q13[i] : i in [1..3] ]); b31 eq 4*(q*p^2-1)*(r+1)*(&+[ a6[i]*p13[i] : i in [1..4] ])+ 4*(r-1)^3*(p+1)*r*(r+1)*(&+[ a4[i]*q13[i] : i in [1..3] ]);