(* This file is for the following artice: Tetsuya ANDO, Some Cubic and Quartic Inequalities of Four Variables. Copy and past suitable part, and test by Mathematica. *) (*===========================================================================*) (*============================== Section 1 ==================================*) (*===========================================================================*) (*--------------------------- Proposition 1.1 -------------------------------*) f13s[a_,b_,c_] := (a^2b+a^2c+b^2a+b^2c+c^2a+c^2b) -6a b c f23s[a_,b_,c_] := a^3+b^3+c^3 + 3 a b c - (a^2b+a^2c+b^2a+b^2c+c^2a+c^2b) f33s[a_,b_,c_] := a b c Factor[f13s[1,1,1]] (* = 0 *) Factor[f13s[1,0,0]] (* = 0 *) Factor[f23s[1,1,1]] (* = 0 *) Factor[f23s[1,1,0]] (* = 0 *) Factor[f33s[1,1,0]] (* = 0 *) Factor[f33s[1,0,0]] (* = 0 *) (*--------------------------- Proposition 1.2 -------------------------------*) S4[a_,b_,c_] := (a^4+b^4+c^4) S31[a_,b_,c_] := (a^3b + b^3c + c^3a) S13[a_,b_,c_] := (a b^3 + b c^3 + c a^3) S22[a_,b_,c_] := (a^2b^2 + b^2c^2 + c^2a^2) US1[a_,b_,c_] := a b c(a + b + c) T31[a_,b_,c_] := S31[a,b,c] + S13[a,b,c] ft4s[a_,b_,c_,t_] := S4[a,b,c] - (t+1) T31[a,b,c] + (t^2+2t) S22[a,b,c] - (t^2-1) US1[a,b,c] Factor[ft4s[t,1,1,t]] (* = 0 *) Factor[ft4s[1,1,1,t]] (* = 0 *) gw[x_,y_,z_,t_] := 2x^2-y^2-z^2 - (t+1)(x y + x z - 2 y z) Factor[6 ft4s[a,b,c,t] - gw[a,b,c,t]^2 - gw[b,c,a,t]^2- gw[c,a,b,t]^2] FrakP[s_,t_] := -(2 S31[s, t, 1] - S13[s, t, 1] - US1[s, t, 1])/(S22[s, t, 1] - US1[s, t, 1]) FrakGX[a_,b_,c_,p_,q_] := S4[a,b,c] + p S31[a,b,c] + q S13[a,b,c] + ((p^2+p q+q^2)/3-1) S22[a,b,c] - (p+q+(p^2+p q+q^2)/3) US1[a,b,c] FrakGA[a_,b_,c_,s_,t_] := FrakGX[a,b,c,FrakP[s,t],FrakP[t,s]] FrakH[a_,b_,c_,s_] := S31[a,b,c] + s^2 S13[a,b,c] - 2s S22[a,b,c] - (s-1)^2 US1[a,b,c] Factor[FrakGA[s,t,1,s,t]] (* = 0 *) Factor[FrakGA[1,1,1,s,t]] (* = 0 *) Factor[FrakGA[a,b,c,1/t,1/t] - ft4s[a,b,c,t]] (* = 0 *) (*----------------------------- Theorem 1.4 ---------------------------------*) S1[a_, b_, c_] := a + b + c S2[a_, b_, c_] := a^2 + b^2 + c^2 S11[a_, b_, c_] := a b + b c + c a S3[a_, b_, c_] := a^3 + b^3 + c^3 S21[a_, b_, c_] := a^2b + b^2c + c^2a S12[a_, b_, c_] := a b^2 + b c^2 + c a^2 T21[a_, b_, c_] := S21[a,b,c] + S12[a,b,c] U[a_, b_, c_] := a b c S4[a_,b_,c_] := (a^4+b^4+c^4) S31[a_,b_,c_] := (a^3b + b^3c + c^3a) S13[a_,b_,c_] := (a b^3 + b c^3 + c a^3) S22[a_,b_,c_] := (a^2b^2 + b^2c^2 + c^2a^2) US1[a_,b_,c_] := a b c(a + b + c) T31[a_,b_,c_] := S31[a,b,c] + S13[a,b,c] S5[a_, b_, c_] := a^5 + b^5 + c^5 S41[a_, b_, c_] := a^4 b + b^4 c + c^4 a S14[a_, b_, c_] := a b^4 + b c^4 + c a^4 S32[a_, b_, c_] := a^3 b^2 + b^3 c^2 + c^3 a^2 S23[a_, b_, c_] := a^2 b^3 + b^2 c^3 + c^2 a^3 T41[a_, b_, c_] := S41[a,b,c] + S14[a,b,c] T32[a_, b_, c_] := S32[a,b,c] + S23[a,b,c] US2[a_, b_, c_] := a b c(a^2+b^2+c^2) US11[a_, b_, c_] := a b c(a b+b c+c a) s0[a_,b_,c_] := S5[a,b,c]-US11[a,b,c] s1[a_,b_,c_] := T41[a,b,c]-2US11[a,b,c] s2[a_,b_,c_] := T32[a,b,c]-2US11[a,b,c] s3[a_,b_,c_] := US2[a,b,c]-US11[a,b,c] s4[a_,b_,c_] := US11[a,b,c] p1A[t_,u_] := (u^2 - (t+2)(5t^2+t+9) u + 9(t-1)^2(t+2)^2)/((5t+1)(t+2)u) p2A[t_,u_] := (-t^2 u^3 + (t-1)(7t^3-t^2+11t+1)u^2 + (t+2)(17t^5-25t^4+199t^3-59t^2+76t+8)u + 9(t-1)^4(t+2)^2(t^2-12t-1))/((5t+1)^3(t+2)u) p3A[t_,u_]:=((2t^3+4t^2+5t+1)u^3 - 2(t+2)(7t^4+42t^3+37t^2+48t+10)u^2 + (t+2)^2(91t^5+125t^4+682t^3+182t^2+523t+125)u - 18(t-1)^2(t+2)^3(t^4+36t^3+34t^2+60t+13))/((t+2)^2(5t+1)^3u) p4A[t_,u_]:=((t-1)^3(6t^2+6t-12 + u)^3)/((t+2)^2(5t+1)^3u) FrakEA[a_,b_,c_,t_,u_] := s0[a,b,c] + p1A[t,u] s1[a,b,c] + p2A[t,u] s2[a,b,c] + p3A[t,u] s3[a,b,c] + p4A[t,u] s4[a,b,c] (* $0 \leq t \leq 7$, $t \ne 1$, $0 \leq u \leq \mu_A(t)$, *) muH[t_] := (t+2)(7-t) muL[t_] := 9 (t-1)^2 muA[t_] := If[t<5/2, muL[t], muH[t]] muZ[t_,u_] := ((t+2)(7-t) - u)/((t+2)(5t+1)) p1B[t_,w_] := -2w-3 p2B[t_,w_] := w^2 + 2w + 2 p3B[t_,w_] := -((2t^3+4t^2+5t+1)/(t^2(t+2))) w^2 + (2(4t^2+5t+3)/(t+2)) w - ((3t^3-7t^2-12t-8)/(t+2)) p4B[t_,w_] := (t-1)^3(-w^2 - 2t^2w +t^2(t-2))/(t^2(t+2)) FrakEB[a_,b_,c_,t_,u_] := s0[a,b,c] + p1B[t,u+1/u-2] s1[a,b,c] + p2B[t,u+1/u-2] s2[a,b,c] + p3B[t,u+1/u-2] s3[a,b,c] + p4B[t,u+1/u-2] s4[a,b,c] (* $t \geq 2$, $s_m(t) \leq u < 1$ *) muR[t_] := 2-t^2 + t Sqrt[(t-1)(t+2)] muB[t_] := (muR[t] - Sqrt[muR[t]^2 - 4])/2 FrakEC[a_,b_,c_,t_] := s0[a,b,c] - (t+1) s1[a,b,c] + t s2[a,b,c] + (t+1)^2 s3[a,b,c] (* $0 \leqq t \leqq 2$ *) FrakED[a_,b_,c_,t_] := s1[a,b,c] + (t^2-1) s2[a,b,c] - 2(t+1)^2 s3[a,b,c] (* $t geqq 0$ *) p3E[t_] := -(4t^2+5t+3)/(t+2) p4E[t_] := (t-1)^3/(t+2) FrakEE[a_,b_,c_,t_] := s1[a,b,c]-s2[a,b,c] + p3E[t] s3[a,b,c] + p4E[t] s4[a,b,c] (*===========================================================================*) (*============================== Section 2 ==================================*) (*===========================================================================*) (*---------------------------- Definition 2.1 -------------------------------*) S4[x0_,x1_,x2_,x3_]:=x0^4+x1^4+x2^4+x3^4 T31[x0_,x1_,x2_,x3_]:=x0^3(x1+x2+x3)+x1^3(x0+x2+x3)+x2^3(x0+x1+x3)+x3^3(x0+x1+x2) S22[x0_,x1_,x2_,x3_]:=x0^2x1^2+x0^2x2^2+x0^2x3^2+x1^2x2^2+x1^2x3^2+x2^2x3^2 T211[x0_,x1_,x2_,x3_]:=x0^2(x1 x2+x1 x3+x2 x3)+x1^2(x0 x2+x0 x3+x2 x3)+x2^2(x0 x1+x0 x3+x1 x3)+x3^2(x0 x1+x0 x2+x1 x2) U[x0_,x1_,x2_,x3_]:=x0 x1 x2 x3 S3[x0_,x1_,x2_,x3_]:=x0^3+x1^3+x2^3+x3^3 S2[x0_,x1_,x2_,x3_]:=x0^2+x1^2+x2^2+x3^2 S1[x0_,x1_,x2_,x3_]:=x0+x1+x2+x3 T21[x0_,x1_,x2_,x3_]:=x0^2(x1+x2+x3)+x1^2(x0+x2+x3)+x2^2(x0+x1+x3)+x3^2(x0+x1+x2) S111[x0_,x1_,x2_,x3_]:=x1 x2 x3 + x0 x2 x3 + x0 x1 x3 + x0 x1 x2 S11[x0_,x1_,x2_,x3_]:=x0 x1+x0 x2+x0 x3+x1 x2+x1 x3+x2 x3 s0[a_,b_,c_,d_] := S4[a,b,c,d] - 4 U[a,b,c,d] s1[a_,b_,c_,d_] := T31[a,b,c,d] - 12 U[a,b,c,d] s2[a_,b_,c_,d_] := S22[a,b,c,d] - 6 U[a,b,c,d] s3[a_,b_,c_,d_] := T211[a,b,c,d] - 12 U[a,b,c,d] s4[a_,b_,c_,d_] := U[a,b,c,d] s[a_,b_,c_,d_] := {s0[a,b,c,d], s1[a,b,c,d], s2[a,b,c,d], s3[a,b,c,d], s4[a,b,c,d]} Omega[u_] := (u-1)^2/u p0G[t_,w_] := (4t+2)w^2 - 3(t-1)^2 w p1G[t_,w_] := -2(t+1)^2w^2 + 2(t+1)(t-1)^2w p2G[t_,w_] := 4t^2w^2 - 2(t-1)^2(2t-1)w + 2(t-1)^4 p3G[t_,w_] := 2(t+1)^2w^2 - (t-1)^2(t^2+3)w - 2(t-1)^4 p4G[t_,w_] := 2(t-1)^4w^2 FrakG[a_,b_,c_,d_,t_,u_] := u^2 (p0G[t,Omega[u]] s0[a,b,c,d] + p1G[t,Omega[u]] s1[a,b,c,d] + p2G[t,Omega[u]] s2[a,b,c,d] + p3G[t,Omega[u]] s3[a,b,c,d] + p4G[t,Omega[u]] s4[a,b,c,d]) (*----------------------------- Theorem 2.2 ---------------------------------*) sa0[a_,b_,c_,d_] := 4 (a^3 - b c d) sa1[a_,b_,c_,d_] := 3 a^2 b + b^3 + 3 a^2 c + c^3 + 3 a^2 d - 12 b c d + d^3 sa2[a_,b_,c_,d_] := 2 (a b^2 + a c^2 - 3 b c d + a d^2) sa3[a_,b_,c_,d_] := 2 a b c + b^2 c + b c^2 + 2 a b d + b^2 d + 2 a c d - 12 b c d + c^2 d + b d^2 + c d^2 sa4[a_,b_,c_,d_] := b c d sa[a_,b_,c_,d_] := {sa0[a,b,c,d], sa1[a,b,c,d], sa2[a,b,c,d], sa3[a,b,c,d], sa4[a,b,c,d]} saa0[a_,b_,c_,d_] := 12a^2 saa1[a_,b_,c_,d_] := 6 a (b + c + d) saa2[a_,b_,c_,d_] := 2 (b^2 + c^2 + d^2) saa3[a_,b_,c_,d_] := 2 (b c + b d + c d) saa[a_,b_,c_,d_] := {saa0[a,b,c,d], saa1[a,b,c,d], saa2[a,b,c,d], saa3[a,b,c,d], saa4[a,b,c,d]} saa4[a_,b_,c_,d_] := 0 saaa0[a_,b_,c_,d_] := 24a saaa1[a_,b_,c_,d_] := 6 (b + c + d) saaa2[a_,b_,c_,d_] := 0 saaa3[a_,b_,c_,d_] := 0 saaa4[a_,b_,c_,d_] := 0 saaa[a_,b_,c_,d_] := {saaa0[a,b,c,d], saaa1[a,b,c,d], saaa2[a,b,c,d], saaa3[a,b,c,d], saaa4[a,b,c,d]} A[t_,u_] :={s[t,1,1,1],sa[t,1,1,1],s[u,u,1,1],sa[u,u,1,1]} B[t_,u_] :={{1,0,0,0,0},s[t,1,1,1],sa[t,1,1,1],s[u,u,1,1],sa[u,u,1,1]} Factor[Det[B[t,u]] - 3(t-1)^2(u^2-1)u^2 p0G[t,u+1/u-2]] Factor[NullSpace[A[t,u]]] Factor[2 (-1 + t)^4 (-1 + u)^4 {-((-2 - 4 t + 7 u + 2 t u + 3 t^2 u - 2 u^2 - 4 t u^2)/( 2 (-1 + t)^4 (-1 + u)^2)), ((1 + t) (-1 - t + 3 u + t^2 u - u^2 - t u^2))/((-1 + t)^4 (-1 + u)^2), ( 1/((-1 + t)^4 (-1 + u)^4))(2 t^2 + u - 4 t u - 3 t^2 u - 2 t^3 u - u^2 + 4 t u^2 + 8 t^2 u^2 + t^4 u^2 + u^3 - 4 t u^3 - 3 t^2 u^3 - 2 t^3 u^3 + 2 t^2 u^4), -(-2 - 4 t - 2 t^2 + 11 u + 10 t u + 12 t^2 u - 2 t^3 u + t^4 u - 16 u^2 - 20 t u^2 - 8 t^2 u^2 - 4 t^3 u^2 + 11 u^3 + 10 t u^3 + 12 t^2 u^3 - 2 t^3 u^3 + t^4 u^3 - 2 u^4 - 4 t u^4 - 2 t^2 u^4)/(2 (-1 + t)^4 (-1 + u)^4), 1}.s[a, b, c, d] - FrG[a, b, c, d, t, u]] (* = 0 *) (*----------------------------- Theorem 2.4 ---------------------------------*) VF[t_,w_] := (3+6t-t^2)w^2 - 6(t-1)^2w Factor[2 p0G[t,w] - VF[t,w]] (* = (t+1)^2w^2 *) (* Thus VF[t,w] \geq 0 ===> p0G[t,w] >= 0 *) Factor[FrakG[-1,0,0,1,t,w] - 2(2(t+1)w-(t-1)^2)^2] (* = 0 *) Factor[FrakG[x,x,1,1,t,w] - 2(t-1)^4(x w -(x-1)^2)^2] (* = 0 *) a1[t_,w_] := (3+4t-t^2)w^2-3(t-1)^2w Factor[FrakG[x,1,1,1,t,w] - (x-t)^2 ((p0G[t,w](x - a1[t,w]/p0G[t,w])^2) + (t-1)^2w^2 VF[t,w]/p0G[t,w])] (*----------------------------- Theorem 2.5 ---------------------------------*) CF[t_,u_] :={ f[t,1,1,1],fa[t,1,1,1],fb[t,1,1,1],fc[t,1,1,1], f[1,t,1,1],fa[1,t,1,1],fb[1,t,1,1],fc[1,t,1,1], f[1,1,t,1],fa[1,1,t,1],fb[1,1,t,1],fc[1,1,t,1], f[1,1,1,t],fa[1,1,1,t],fb[1,1,1,t],fc[1,1,1,t], f[u,u,1,1],fa[u,u,1,1],fb[u,u,1,1],fc[u,u,1,1], f[u,1,u,1],fa[u,1,u,1],fb[u,1,u,1],fc[u,1,u,1], f[u,1,1,u],fa[u,1,1,u],fb[u,1,1,u],fc[u,1,1,u], f[1,u,u,1],fa[1,u,u,1],fb[1,u,u,1],fc[1,u,u,1], f[1,u,1,u],fa[1,u,1,u],fb[1,u,1,u],fc[1,u,1,u], f[1,1,u,u],fa[1,1,u,u],fb[1,1,u,u],fc[1,1,u,u]} e1 := {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} Factor[Det[{e1, f[t,1,1,1],fa[t,1,1,1],fb[t,1,1,1],fc[t,1,1,1], f[1,t,1,1],fa[1,t,1,1],fb[1,t,1,1],fc[1,t,1,1], f[1,1,t,1],fa[1,1,t,1],fb[1,1,t,1],fc[1,1,t,1], f[1,1,1,t],fa[1,1,1,t],fb[1,1,1,t],fc[1,1,1,t], f[u,u,1,1],fa[u,u,1,1],fb[u,u,1,1],fc[u,u,1,1], f[u,1,u,1],fa[u,1,u,1],fb[u,1,u,1],fc[u,1,u,1], f[u,1,1,u],fa[u,1,1,u],fb[u,1,1,u],fc[u,1,1,u], f[1,u,u,1],fa[1,u,u,1],fb[1,u,u,1], f[1,u,1,u],fa[1,u,1,u], f[1,1,u,u]}]] dFrG[t_,u_] := -(-1 + t)^29 t (1 + t - 2 u)^4 (-1 + u)^31 u (1 + u)^9 (-2 + u + t u)^3 (-2 - 4 t + 7 u + 2 t u + 3 t^2 u - 2 u^2 - 4 t u^2) (-3 - 6 t + t^2 + 12 u + 4 t^2 u - 3 u^2 - 6 t u^2 + t^2 u^2) (3 + 6 t - t^2 - 13 u - 6 t u - 5 t^2 u + 10 u^2 + 9 t u^2 + 4 t^2 u^2 + t^3 u^2 - u^3 - 6 t u^3 - t^2 u^3)^2 VF[t_,w_] := (3+6t-t^2)w^2 - 6(t-1)^2w DF[t_,u_] := (3+6t-t^2) - (13+6t+5t^2)u + (10+9t+4t^2+t^3)u^2 - (1+6t+t^2)u^3 Factor[dFrG[t,u] + t(t-1)^(29) u^5(u-1)^(27)(u+1)^9 (t-2u+1)^4(t u + u - 2)^3 p0G[t,u+1/u-2] VF[t,u+1/u-2] DF[t,u]^2] VFw[t_,w_] := (3+6t-t^2)w - 6(t-1)^2 Omega[u_] := (u-1)^2/u ContourPlot[{VFw[t,Omega[u]]==0, DF[t,u]==0, u==-1, u==0, u==1, 2u==t+1, t u+u=2}, {t,-5,5},{u,-5,5}] (*----------------------------- Theorem 2.6 ---------------------------------*) q1[a_,b_,c_,d_]:=a^2 q2[a_,b_,c_,d_]:=a b q3[a_,b_,c_,d_]:=a c q4[a_,b_,c_,d_]:=a d q5[a_,b_,c_,d_]:=b^2 q6[a_,b_,c_,d_]:=b c q7[a_,b_,c_,d_]:=b d q8[a_,b_,c_,d_]:=c^2 q9[a_,b_,c_,d_]:=c d q10[a_,b_,c_,d_]:=d^2 q[a_,b_,c_,d_]:={q1[a,b,c,d],q2[a,b,c,d],q3[a,b,c,d],q4[a,b,c,d],q5[a,b,c,d],q6[a,b,c,d],q7[a,b,c,d],q8[a,b,c,d],q9[a,b,c,d],q10[a,b,c,d]} (* Zero points of $\frf_{t,u}$ *) AQ[t_,u_] := {q[t,1,1,1], q[1,t,1,1], q[1,1,t,1], q[1,1,1,t], q[u,u,1,1], q[u,1,u,1], q[u,1,1,u], q[1,u,u,1], q[1,u,1,u], q[1,1,u,u]} Factor[Det[AQ[t,u]] + (t-1)^6(u-1)^5(u+1)^3 u^2 VF[t,u+1/u-2]] (* = 0 *) (*============================= Section 2.2 =================================*) (*--------------------- Proof of Ptoposition 1.6 ----------------------------*) S3[a_,b_,c_,d_] := a^3+b^3+c^3+d^3 T21[a_,b_,c_,d_] := a^2b+a^2c+a^2d+b^2a+b^2c+b^2d+c^2a+c^2b+c^2d+d^2a+d^2b+d^2c S111[a_,b_,c_,d_] := b c d + a c d + a b d + a b c s0[a_,b_,c_,d_] := S3[a,b,c,d] - S111[a,b,c,d] s1[a_,b_,c_,d_] := T21[a,b,c,d] - 3 S111[a,b,c,d] s2[a_,b_,c_,d_] := S111[a,b,c,d] Phi[a_,b_,c_,d_] := {s0[a,b,c,d], s1[a,b,c,d], s2[a,b,c,d]} g13s[a_,b_,c_,d_] := s1[a,b,c,d] g23s[a_,b_,c_,d_] := 3s0[a,b,c,d] - 2 s1[a,b,c,d] g33s[a_,b_,c_,d_] := s2[a,b,c,d] g43s[a_,b_,c_,d_] := s0[a,b,c,d] - s1[a,b,c,d] + s2[a,b,c,d] (* (dual of $P_1P_2$) *) Factor[g13s[0,0,0,1]] (* = 0 *) Factor[g13s[1,1,1,1]] (* = 0 *) Factor[g23s[0,0,1,1]] (* = 0 *) Factor[g23s[1,1,1,1]] (* = 0 *) Factor[g33s[0,0,0,1]] (* = 0 *) Factor[g33s[0,0,1,1]] (* = 0 *) Factor[g43s[0,0,1,1]] (* = 0 *) Factor[g43s[0,1,1,1]] (* = 0 *) f13s[a_,b_,c_] := (a^2b+a^2c+b^2a+b^2c+c^2a+c^2b) -6a b c f23s[a_,b_,c_] := a^3+b^3+c^3 + 3 a b c - (a^2b+a^2c+b^2a+b^2c+c^2a+c^2b) f33s[a_,b_,c_] := a b c Factor[(f13s[b,c,d] + f13s[a,c,d] + f13s[a,b,d] + f13s[a,b,c]) - 2 g13s[a,b,c,d]] (* = 0 *) Factor[(f23s[b,c,d] + f23s[a,c,d] + f23s[a,b,d] + f23s[a,b,c]) - g23s[a,b,c,d]] (* = 0 *) Factor[(f33s[b,c,d] + f33s[a,c,d] + f33s[a,b,d] + f33s[a,b,c]) - g33s[a,b,c,d]] (* = 0 *) ParametricPlot[{ {s1[0,0,x,1]/s0[0,0,x,1], 0}, {s1[0,x,x,1]/s0[0,x,x,1], s2[0,x,x,1]/s0[0,x,x,1]}, {3x/(2x+1), s2[x,x,x,1]/s0[x,x,x,1]}, {3/(x+2), s2[x,1,1,1]/s0[x,1,1,1]}, {1, s2[x,x,1,1]/s0[x,x,1,1]}, {s1[0,x,1,1]/s0[0,x,1,1], s2[0,x,1,1]/s0[0,x,1,1]}}, {x,0,0.9}] Eliminate[{x==s1[t,t,0,1]/s0[t,t,0,1], y==s2[t,t,0,1]/s0[t,t,0,1]}, t] Eliminate[{x==s1[t,0,1,1]/s0[t,0,1,1], y==s2[t,0,1,1]/s0[t,0,1,1]}, t] (* -x^2 + x^3 + 4y - 6 x y + 2x^2y + 11y^2 - 15 x y^2 + 16y^3 == 0 Cusp (x,y)=(3/2, 1/2). *) Eliminate[{x==3t/(2t+1), y==s2[t,t,t,1]/s0[t,t,t,1]}, t] Eliminate[{x==3/(t+2), y==s2[t,1,1,1]/s0[t,1,1,1]}, t] (* y == x^2(9-5x)/(27(x-1)^2) *) (*--------------------------- Theorem 2.10 (1) ------------------------------*) f1[a_,b_,c_,d_]:=a^6 f2[a_,b_,c_,d_]:=a^5b f3[a_,b_,c_,d_]:=a^5c f4[a_,b_,c_,d_]:=a^5d f5[a_,b_,c_,d_]:=a^4b^2 f6[a_,b_,c_,d_]:=a^4b c f7[a_,b_,c_,d_]:=a^4b d f8[a_,b_,c_,d_]:=a^4c^2 f9[a_,b_,c_,d_]:=a^4c d f10[a_,b_,c_,d_]:=a^4d^2 f11[a_,b_,c_,d_]:=a^3b^3 f12[a_,b_,c_,d_]:=a^3b^2c f13[a_,b_,c_,d_]:=a^3b^2d f14[a_,b_,c_,d_]:=a^3b c^2 f15[a_,b_,c_,d_]:=a^3b c d f16[a_,b_,c_,d_]:=a^3b d^2 f17[a_,b_,c_,d_]:=a^3c^3 f18[a_,b_,c_,d_]:=a^3c^2d f19[a_,b_,c_,d_]:=a^3c d^2 f20[a_,b_,c_,d_]:=a^3d^3 f21[a_,b_,c_,d_]:=a^2b^4 f22[a_,b_,c_,d_]:=a^2b^3c f23[a_,b_,c_,d_]:=a^2b^3d f24[a_,b_,c_,d_]:=a^2b^2c^2 f25[a_,b_,c_,d_]:=a^2b^2c d f26[a_,b_,c_,d_]:=a^2b^2d^2 f27[a_,b_,c_,d_]:=a^2b c^3 f28[a_,b_,c_,d_]:=a^2b c^2d f29[a_,b_,c_,d_]:=a^2b c d^2 f30[a_,b_,c_,d_]:=a^2b d^3 f31[a_,b_,c_,d_]:=a^2c^4 f32[a_,b_,c_,d_]:=a^2c^3d f33[a_,b_,c_,d_]:=a^2c^2d^2 f34[a_,b_,c_,d_]:=a^2c d^3 f35[a_,b_,c_,d_]:=a^2d^4 f36[a_,b_,c_,d_]:=a b^5 f37[a_,b_,c_,d_]:=a b^4c f38[a_,b_,c_,d_]:=a b^4d f39[a_,b_,c_,d_]:=a b^3c^2 f40[a_,b_,c_,d_]:=a b^3c d f41[a_,b_,c_,d_]:=a b^3d^2 f42[a_,b_,c_,d_]:=a b^2c^3 f43[a_,b_,c_,d_]:=a b^2c^2d f44[a_,b_,c_,d_]:=a b^2c d^2 f45[a_,b_,c_,d_]:=a b^2d^3 f46[a_,b_,c_,d_]:=a b c^4 f47[a_,b_,c_,d_]:=a b c^3d f48[a_,b_,c_,d_]:=a b c^2d^2 f49[a_,b_,c_,d_]:=a b c d^3 f50[a_,b_,c_,d_]:=a b d^4 f51[a_,b_,c_,d_]:=a c^5 f52[a_,b_,c_,d_]:=a c^4d f53[a_,b_,c_,d_]:=a c^3d^2 f54[a_,b_,c_,d_]:=a c^2d^3 f55[a_,b_,c_,d_]:=a c d^4 f56[a_,b_,c_,d_]:=a d^5 f57[a_,b_,c_,d_]:=b^6 f58[a_,b_,c_,d_]:=b^5c f59[a_,b_,c_,d_]:=b^5d f60[a_,b_,c_,d_]:=b^4c^2 f61[a_,b_,c_,d_]:=b^4c d f62[a_,b_,c_,d_]:=b^4d^2 f63[a_,b_,c_,d_]:=b^3c^3 f64[a_,b_,c_,d_]:=b^3c^2d f65[a_,b_,c_,d_]:=b^3c d^2 f66[a_,b_,c_,d_]:=b^3d^3 f67[a_,b_,c_,d_]:=b^2c^4 f68[a_,b_,c_,d_]:=b^2c^3d f69[a_,b_,c_,d_]:=b^2c^2d^2 f70[a_,b_,c_,d_]:=b^2c d^3 f71[a_,b_,c_,d_]:=b^2d^4 f72[a_,b_,c_,d_]:=b c^5 f73[a_,b_,c_,d_]:=b c^4d f74[a_,b_,c_,d_]:=b c^3d^2 f75[a_,b_,c_,d_]:=b c^2d^3 f76[a_,b_,c_,d_]:=b c d^4 f77[a_,b_,c_,d_]:=b d^5 f78[a_,b_,c_,d_]:=c^6 f79[a_,b_,c_,d_]:=c^5d f80[a_,b_,c_,d_]:=c^4d^2 f81[a_,b_,c_,d_]:=c^3d^3 f82[a_,b_,c_,d_]:=c^2d^4 f83[a_,b_,c_,d_]:=c d^5 f84[a_,b_,c_,d_]:=d^6 f[a_,b_,c_,d_]:={f1[a,b,c,d],f2[a,b,c,d],f3[a,b,c,d],f4[a,b,c,d],f5[a,b,c,d],f6[a,b,c,d],f7[a,b,c,d],f8[a,b,c,d],f9[a,b,c,d],f10[a,b,c,d],f11[a,b,c,d],f12[a,b,c,d],f13[a,b,c,d],f14[a,b,c,d],f15[a,b,c,d],f16[a,b,c,d],f17[a,b,c,d],f18[a,b,c,d],f19[a,b,c,d],f20[a,b,c,d],f21[a,b,c,d],f22[a,b,c,d],f23[a,b,c,d],f24[a,b,c,d],f25[a,b,c,d],f26[a,b,c,d],f27[a,b,c,d],f28[a,b,c,d],f29[a,b,c,d],f30[a,b,c,d],f31[a,b,c,d],f32[a,b,c,d],f33[a,b,c,d],f34[a,b,c,d],f35[a,b,c,d],f36[a,b,c,d],f37[a,b,c,d],f38[a,b,c,d],f39[a,b,c,d],f40[a,b,c,d],f41[a,b,c,d],f42[a,b,c,d],f43[a,b,c,d],f44[a,b,c,d],f45[a,b,c,d],f46[a,b,c,d],f47[a,b,c,d],f48[a,b,c,d],f49[a,b,c,d],f50[a,b,c,d],f51[a,b,c,d],f52[a,b,c,d],f53[a,b,c,d],f54[a,b,c,d],f55[a,b,c,d],f56[a,b,c,d],f57[a,b,c,d],f58[a,b,c,d],f59[a,b,c,d],f60[a,b,c,d],f61[a,b,c,d],f62[a,b,c,d],f63[a,b,c,d],f64[a,b,c,d],f65[a,b,c,d],f66[a,b,c,d],f67[a,b,c,d],f68[a,b,c,d],f69[a,b,c,d],f70[a,b,c,d],f71[a,b,c,d],f72[a,b,c,d],f73[a,b,c,d],f74[a,b,c,d],f75[a,b,c,d],f76[a,b,c,d],f77[a,b,c,d],f78[a,b,c,d],f79[a,b,c,d],f80[a,b,c,d],f81[a,b,c,d],f82[a,b,c,d],f83[a,b,c,d],f84[a,b,c,d]} fa1[a_,b_,c_,d_]:=6a^5 fa2[a_,b_,c_,d_]:=5a^4b fa3[a_,b_,c_,d_]:=5a^4c fa4[a_,b_,c_,d_]:=5a^4d fa5[a_,b_,c_,d_]:=4a^3b^2 fa6[a_,b_,c_,d_]:=4a^3b c fa7[a_,b_,c_,d_]:=4a^3b d fa8[a_,b_,c_,d_]:=4a^3c^2 fa9[a_,b_,c_,d_]:=4a^3c d fa10[a_,b_,c_,d_]:=4a^3d^2 fa11[a_,b_,c_,d_]:=3a^2b^3 fa12[a_,b_,c_,d_]:=3a^2b^2c fa13[a_,b_,c_,d_]:=3a^2b^2d fa14[a_,b_,c_,d_]:=3a^2b c^2 fa15[a_,b_,c_,d_]:=3a^2b c d fa16[a_,b_,c_,d_]:=3a^2b d^2 fa17[a_,b_,c_,d_]:=3a^2c^3 fa18[a_,b_,c_,d_]:=3a^2c^2d fa19[a_,b_,c_,d_]:=3a^2c d^2 fa20[a_,b_,c_,d_]:=3a^2d^3 fa21[a_,b_,c_,d_]:=2a b^4 fa22[a_,b_,c_,d_]:=2a b^3c fa23[a_,b_,c_,d_]:=2a b^3d fa24[a_,b_,c_,d_]:=2a b^2c^2 fa25[a_,b_,c_,d_]:=2a b^2c d fa26[a_,b_,c_,d_]:=2a b^2d^2 fa27[a_,b_,c_,d_]:=2a b c^3 fa28[a_,b_,c_,d_]:=2a b c^2d fa29[a_,b_,c_,d_]:=2a b c d^2 fa30[a_,b_,c_,d_]:=2a b d^3 fa31[a_,b_,c_,d_]:=2a c^4 fa32[a_,b_,c_,d_]:=2a c^3d fa33[a_,b_,c_,d_]:=2a c^2d^2 fa34[a_,b_,c_,d_]:=2a c d^3 fa35[a_,b_,c_,d_]:=2a d^4 fa36[a_,b_,c_,d_]:=b^5 fa37[a_,b_,c_,d_]:=b^4c fa38[a_,b_,c_,d_]:=b^4d fa39[a_,b_,c_,d_]:=b^3c^2 fa40[a_,b_,c_,d_]:=b^3c d fa41[a_,b_,c_,d_]:=b^3d^2 fa42[a_,b_,c_,d_]:=b^2c^3 fa43[a_,b_,c_,d_]:=b^2c^2d fa44[a_,b_,c_,d_]:=b^2c d^2 fa45[a_,b_,c_,d_]:=b^2d^3 fa46[a_,b_,c_,d_]:=b c^4 fa47[a_,b_,c_,d_]:=b c^3d fa48[a_,b_,c_,d_]:=b c^2d^2 fa49[a_,b_,c_,d_]:=b c d^3 fa50[a_,b_,c_,d_]:=b d^4 fa51[a_,b_,c_,d_]:=c^5 fa52[a_,b_,c_,d_]:=c^4d fa53[a_,b_,c_,d_]:=c^3d^2 fa54[a_,b_,c_,d_]:=c^2d^3 fa55[a_,b_,c_,d_]:=c d^4 fa56[a_,b_,c_,d_]:=d^5 fa57[a_,b_,c_,d_]:=0 fa58[a_,b_,c_,d_]:=0 fa59[a_,b_,c_,d_]:=0 fa60[a_,b_,c_,d_]:=0 fa61[a_,b_,c_,d_]:=0 fa62[a_,b_,c_,d_]:=0 fa63[a_,b_,c_,d_]:=0 fa64[a_,b_,c_,d_]:=0 fa65[a_,b_,c_,d_]:=0 fa66[a_,b_,c_,d_]:=0 fa67[a_,b_,c_,d_]:=0 fa68[a_,b_,c_,d_]:=0 fa69[a_,b_,c_,d_]:=0 fa70[a_,b_,c_,d_]:=0 fa71[a_,b_,c_,d_]:=0 fa72[a_,b_,c_,d_]:=0 fa73[a_,b_,c_,d_]:=0 fa74[a_,b_,c_,d_]:=0 fa75[a_,b_,c_,d_]:=0 fa76[a_,b_,c_,d_]:=0 fa77[a_,b_,c_,d_]:=0 fa78[a_,b_,c_,d_]:=0 fa79[a_,b_,c_,d_]:=0 fa80[a_,b_,c_,d_]:=0 fa81[a_,b_,c_,d_]:=0 fa82[a_,b_,c_,d_]:=0 fa83[a_,b_,c_,d_]:=0 fa84[a_,b_,c_,d_]:=0 fa[a_,b_,c_,d_]:={fa1[a,b,c,d],fa2[a,b,c,d],fa3[a,b,c,d],fa4[a,b,c,d],fa5[a,b,c,d],fa6[a,b,c,d],fa7[a,b,c,d],fa8[a,b,c,d],fa9[a,b,c,d],fa10[a,b,c,d],fa11[a,b,c,d],fa12[a,b,c,d],fa13[a,b,c,d],fa14[a,b,c,d],fa15[a,b,c,d],fa16[a,b,c,d],fa17[a,b,c,d],fa18[a,b,c,d],fa19[a,b,c,d],fa20[a,b,c,d],fa21[a,b,c,d],fa22[a,b,c,d],fa23[a,b,c,d],fa24[a,b,c,d],fa25[a,b,c,d],fa26[a,b,c,d],fa27[a,b,c,d],fa28[a,b,c,d],fa29[a,b,c,d],fa30[a,b,c,d],fa31[a,b,c,d],fa32[a,b,c,d],fa33[a,b,c,d],fa34[a,b,c,d],fa35[a,b,c,d],fa36[a,b,c,d],fa37[a,b,c,d],fa38[a,b,c,d],fa39[a,b,c,d],fa40[a,b,c,d],fa41[a,b,c,d],fa42[a,b,c,d],fa43[a,b,c,d],fa44[a,b,c,d],fa45[a,b,c,d],fa46[a,b,c,d],fa47[a,b,c,d],fa48[a,b,c,d],fa49[a,b,c,d],fa50[a,b,c,d],fa51[a,b,c,d],fa52[a,b,c,d],fa53[a,b,c,d],fa54[a,b,c,d],fa55[a,b,c,d],fa56[a,b,c,d],fa57[a,b,c,d],fa58[a,b,c,d],fa59[a,b,c,d],fa60[a,b,c,d],fa61[a,b,c,d],fa62[a,b,c,d],fa63[a,b,c,d],fa64[a,b,c,d],fa65[a,b,c,d],fa66[a,b,c,d],fa67[a,b,c,d],fa68[a,b,c,d],fa69[a,b,c,d],fa70[a,b,c,d],fa71[a,b,c,d],fa72[a,b,c,d],fa73[a,b,c,d],fa74[a,b,c,d],fa75[a,b,c,d],fa76[a,b,c,d],fa77[a,b,c,d],fa78[a,b,c,d],fa79[a,b,c,d],fa80[a,b,c,d],fa81[a,b,c,d],fa82[a,b,c,d],fa83[a,b,c,d],fa84[a,b,c,d]} fb1[a_,b_,c_,d_]:=0 fb2[a_,b_,c_,d_]:=a^5 fb3[a_,b_,c_,d_]:=0 fb4[a_,b_,c_,d_]:=0 fb5[a_,b_,c_,d_]:=2a^4b fb6[a_,b_,c_,d_]:=a^4c fb7[a_,b_,c_,d_]:=a^4d fb8[a_,b_,c_,d_]:=0 fb9[a_,b_,c_,d_]:=0 fb10[a_,b_,c_,d_]:=0 fb11[a_,b_,c_,d_]:=3a^3b^2 fb12[a_,b_,c_,d_]:=2a^3b c fb13[a_,b_,c_,d_]:=2a^3b d fb14[a_,b_,c_,d_]:=a^3c^2 fb15[a_,b_,c_,d_]:=a^3c d fb16[a_,b_,c_,d_]:=a^3d^2 fb17[a_,b_,c_,d_]:=0 fb18[a_,b_,c_,d_]:=0 fb19[a_,b_,c_,d_]:=0 fb20[a_,b_,c_,d_]:=0 fb21[a_,b_,c_,d_]:=4a^2b^3 fb22[a_,b_,c_,d_]:=3a^2b^2c fb23[a_,b_,c_,d_]:=3a^2b^2d fb24[a_,b_,c_,d_]:=2a^2b c^2 fb25[a_,b_,c_,d_]:=2a^2b c d fb26[a_,b_,c_,d_]:=2a^2b d^2 fb27[a_,b_,c_,d_]:=a^2c^3 fb28[a_,b_,c_,d_]:=a^2c^2d fb29[a_,b_,c_,d_]:=a^2c d^2 fb30[a_,b_,c_,d_]:=a^2d^3 fb31[a_,b_,c_,d_]:=0 fb32[a_,b_,c_,d_]:=0 fb33[a_,b_,c_,d_]:=0 fb34[a_,b_,c_,d_]:=0 fb35[a_,b_,c_,d_]:=0 fb36[a_,b_,c_,d_]:=5a b^4 fb37[a_,b_,c_,d_]:=4a b^3c fb38[a_,b_,c_,d_]:=4a b^3d fb39[a_,b_,c_,d_]:=3a b^2c^2 fb40[a_,b_,c_,d_]:=3a b^2c d fb41[a_,b_,c_,d_]:=3a b^2d^2 fb42[a_,b_,c_,d_]:=2a b c^3 fb43[a_,b_,c_,d_]:=2a b c^2d fb44[a_,b_,c_,d_]:=2a b c d^2 fb45[a_,b_,c_,d_]:=2a b d^3 fb46[a_,b_,c_,d_]:=a c^4 fb47[a_,b_,c_,d_]:=a c^3d fb48[a_,b_,c_,d_]:=a c^2d^2 fb49[a_,b_,c_,d_]:=a c d^3 fb50[a_,b_,c_,d_]:=a d^4 fb51[a_,b_,c_,d_]:=0 fb52[a_,b_,c_,d_]:=0 fb53[a_,b_,c_,d_]:=0 fb54[a_,b_,c_,d_]:=0 fb55[a_,b_,c_,d_]:=0 fb56[a_,b_,c_,d_]:=0 fb57[a_,b_,c_,d_]:=6b^5 fb58[a_,b_,c_,d_]:=5b^4c fb59[a_,b_,c_,d_]:=5b^4d fb60[a_,b_,c_,d_]:=4b^3c^2 fb61[a_,b_,c_,d_]:=4b^3c d fb62[a_,b_,c_,d_]:=4b^3d^2 fb63[a_,b_,c_,d_]:=3b^2c^3 fb64[a_,b_,c_,d_]:=3b^2c^2d fb65[a_,b_,c_,d_]:=3b^2c d^2 fb66[a_,b_,c_,d_]:=3b^2d^3 fb67[a_,b_,c_,d_]:=2b c^4 fb68[a_,b_,c_,d_]:=2b c^3d fb69[a_,b_,c_,d_]:=2b c^2d^2 fb70[a_,b_,c_,d_]:=2b c d^3 fb71[a_,b_,c_,d_]:=2b d^4 fb72[a_,b_,c_,d_]:=c^5 fb73[a_,b_,c_,d_]:=c^4d fb74[a_,b_,c_,d_]:=c^3d^2 fb75[a_,b_,c_,d_]:=c^2d^3 fb76[a_,b_,c_,d_]:=c d^4 fb77[a_,b_,c_,d_]:=d^5 fb78[a_,b_,c_,d_]:=0 fb79[a_,b_,c_,d_]:=0 fb80[a_,b_,c_,d_]:=0 fb81[a_,b_,c_,d_]:=0 fb82[a_,b_,c_,d_]:=0 fb83[a_,b_,c_,d_]:=0 fb84[a_,b_,c_,d_]:=0 fb[a_,b_,c_,d_]:={fb1[a,b,c,d],fb2[a,b,c,d],fb3[a,b,c,d],fb4[a,b,c,d],fb5[a,b,c,d],fb6[a,b,c,d],fb7[a,b,c,d],fb8[a,b,c,d],fb9[a,b,c,d],fb10[a,b,c,d],fb11[a,b,c,d],fb12[a,b,c,d],fb13[a,b,c,d],fb14[a,b,c,d],fb15[a,b,c,d],fb16[a,b,c,d],fb17[a,b,c,d],fb18[a,b,c,d],fb19[a,b,c,d],fb20[a,b,c,d],fb21[a,b,c,d],fb22[a,b,c,d],fb23[a,b,c,d],fb24[a,b,c,d],fb25[a,b,c,d],fb26[a,b,c,d],fb27[a,b,c,d],fb28[a,b,c,d],fb29[a,b,c,d],fb30[a,b,c,d],fb31[a,b,c,d],fb32[a,b,c,d],fb33[a,b,c,d],fb34[a,b,c,d],fb35[a,b,c,d],fb36[a,b,c,d],fb37[a,b,c,d],fb38[a,b,c,d],fb39[a,b,c,d],fb40[a,b,c,d],fb41[a,b,c,d],fb42[a,b,c,d],fb43[a,b,c,d],fb44[a,b,c,d],fb45[a,b,c,d],fb46[a,b,c,d],fb47[a,b,c,d],fb48[a,b,c,d],fb49[a,b,c,d],fb50[a,b,c,d],fb51[a,b,c,d],fb52[a,b,c,d],fb53[a,b,c,d],fb54[a,b,c,d],fb55[a,b,c,d],fb56[a,b,c,d],fb57[a,b,c,d],fb58[a,b,c,d],fb59[a,b,c,d],fb60[a,b,c,d],fb61[a,b,c,d],fb62[a,b,c,d],fb63[a,b,c,d],fb64[a,b,c,d],fb65[a,b,c,d],fb66[a,b,c,d],fb67[a,b,c,d],fb68[a,b,c,d],fb69[a,b,c,d],fb70[a,b,c,d],fb71[a,b,c,d],fb72[a,b,c,d],fb73[a,b,c,d],fb74[a,b,c,d],fb75[a,b,c,d],fb76[a,b,c,d],fb77[a,b,c,d],fb78[a,b,c,d],fb79[a,b,c,d],fb80[a,b,c,d],fb81[a,b,c,d],fb82[a,b,c,d],fb83[a,b,c,d],fb84[a,b,c,d]} fc1[a_,b_,c_,d_]:=0 fc2[a_,b_,c_,d_]:=0 fc3[a_,b_,c_,d_]:=a^5 fc4[a_,b_,c_,d_]:=0 fc5[a_,b_,c_,d_]:=0 fc6[a_,b_,c_,d_]:=a^4b fc7[a_,b_,c_,d_]:=0 fc8[a_,b_,c_,d_]:=2a^4c fc9[a_,b_,c_,d_]:=a^4d fc10[a_,b_,c_,d_]:=0 fc11[a_,b_,c_,d_]:=0 fc12[a_,b_,c_,d_]:=a^3b^2 fc13[a_,b_,c_,d_]:=0 fc14[a_,b_,c_,d_]:=2a^3b c fc15[a_,b_,c_,d_]:=a^3b d fc16[a_,b_,c_,d_]:=0 fc17[a_,b_,c_,d_]:=3a^3c^2 fc18[a_,b_,c_,d_]:=2a^3c d fc19[a_,b_,c_,d_]:=a^3d^2 fc20[a_,b_,c_,d_]:=0 fc21[a_,b_,c_,d_]:=0 fc22[a_,b_,c_,d_]:=a^2b^3 fc23[a_,b_,c_,d_]:=0 fc24[a_,b_,c_,d_]:=2a^2b^2c fc25[a_,b_,c_,d_]:=a^2b^2d fc26[a_,b_,c_,d_]:=0 fc27[a_,b_,c_,d_]:=3a^2b c^2 fc28[a_,b_,c_,d_]:=2a^2b c d fc29[a_,b_,c_,d_]:=a^2b d^2 fc30[a_,b_,c_,d_]:=0 fc31[a_,b_,c_,d_]:=4a^2c^3 fc32[a_,b_,c_,d_]:=3a^2c^2d fc33[a_,b_,c_,d_]:=2a^2c d^2 fc34[a_,b_,c_,d_]:=a^2d^3 fc35[a_,b_,c_,d_]:=0 fc36[a_,b_,c_,d_]:=0 fc37[a_,b_,c_,d_]:=a b^4 fc38[a_,b_,c_,d_]:=0 fc39[a_,b_,c_,d_]:=2a b^3c fc40[a_,b_,c_,d_]:=a b^3d fc41[a_,b_,c_,d_]:=0 fc42[a_,b_,c_,d_]:=3a b^2c^2 fc43[a_,b_,c_,d_]:=2a b^2c d fc44[a_,b_,c_,d_]:=a b^2d^2 fc45[a_,b_,c_,d_]:=0 fc46[a_,b_,c_,d_]:=4a b c^3 fc47[a_,b_,c_,d_]:=3a b c^2d fc48[a_,b_,c_,d_]:=2a b c d^2 fc49[a_,b_,c_,d_]:=a b d^3 fc50[a_,b_,c_,d_]:=0 fc51[a_,b_,c_,d_]:=5a c^4 fc52[a_,b_,c_,d_]:=4a c^3d fc53[a_,b_,c_,d_]:=3a c^2d^2 fc54[a_,b_,c_,d_]:=2a c d^3 fc55[a_,b_,c_,d_]:=a d^4 fc56[a_,b_,c_,d_]:=0 fc57[a_,b_,c_,d_]:=0 fc58[a_,b_,c_,d_]:=b^5 fc59[a_,b_,c_,d_]:=0 fc60[a_,b_,c_,d_]:=2b^4c fc61[a_,b_,c_,d_]:=b^4d fc62[a_,b_,c_,d_]:=0 fc63[a_,b_,c_,d_]:=3b^3c^2 fc64[a_,b_,c_,d_]:=2b^3c d fc65[a_,b_,c_,d_]:=b^3d^2 fc66[a_,b_,c_,d_]:=0 fc67[a_,b_,c_,d_]:=4b^2c^3 fc68[a_,b_,c_,d_]:=3b^2c^2d fc69[a_,b_,c_,d_]:=2b^2c d^2 fc70[a_,b_,c_,d_]:=b^2d^3 fc71[a_,b_,c_,d_]:=0 fc72[a_,b_,c_,d_]:=5b c^4 fc73[a_,b_,c_,d_]:=4b c^3d fc74[a_,b_,c_,d_]:=3b c^2d^2 fc75[a_,b_,c_,d_]:=2b c d^3 fc76[a_,b_,c_,d_]:=b d^4 fc77[a_,b_,c_,d_]:=0 fc78[a_,b_,c_,d_]:=6c^5 fc79[a_,b_,c_,d_]:=5c^4d fc80[a_,b_,c_,d_]:=4c^3d^2 fc81[a_,b_,c_,d_]:=3c^2d^3 fc82[a_,b_,c_,d_]:=2c d^4 fc83[a_,b_,c_,d_]:=d^5 fc84[a_,b_,c_,d_]:=0 fc[a_,b_,c_,d_]:={fc1[a,b,c,d],fc2[a,b,c,d],fc3[a,b,c,d],fc4[a,b,c,d],fc5[a,b,c,d],fc6[a,b,c,d],fc7[a,b,c,d],fc8[a,b,c,d],fc9[a,b,c,d],fc10[a,b,c,d],fc11[a,b,c,d],fc12[a,b,c,d],fc13[a,b,c,d],fc14[a,b,c,d],fc15[a,b,c,d],fc16[a,b,c,d],fc17[a,b,c,d],fc18[a,b,c,d],fc19[a,b,c,d],fc20[a,b,c,d],fc21[a,b,c,d],fc22[a,b,c,d],fc23[a,b,c,d],fc24[a,b,c,d],fc25[a,b,c,d],fc26[a,b,c,d],fc27[a,b,c,d],fc28[a,b,c,d],fc29[a,b,c,d],fc30[a,b,c,d],fc31[a,b,c,d],fc32[a,b,c,d],fc33[a,b,c,d],fc34[a,b,c,d],fc35[a,b,c,d],fc36[a,b,c,d],fc37[a,b,c,d],fc38[a,b,c,d],fc39[a,b,c,d],fc40[a,b,c,d],fc41[a,b,c,d],fc42[a,b,c,d],fc43[a,b,c,d],fc44[a,b,c,d],fc45[a,b,c,d],fc46[a,b,c,d],fc47[a,b,c,d],fc48[a,b,c,d],fc49[a,b,c,d],fc50[a,b,c,d],fc51[a,b,c,d],fc52[a,b,c,d],fc53[a,b,c,d],fc54[a,b,c,d],fc55[a,b,c,d],fc56[a,b,c,d],fc57[a,b,c,d],fc58[a,b,c,d],fc59[a,b,c,d],fc60[a,b,c,d],fc61[a,b,c,d],fc62[a,b,c,d],fc63[a,b,c,d],fc64[a,b,c,d],fc65[a,b,c,d],fc66[a,b,c,d],fc67[a,b,c,d],fc68[a,b,c,d],fc69[a,b,c,d],fc70[a,b,c,d],fc71[a,b,c,d],fc72[a,b,c,d],fc73[a,b,c,d],fc74[a,b,c,d],fc75[a,b,c,d],fc76[a,b,c,d],fc77[a,b,c,d],fc78[a,b,c,d],fc79[a,b,c,d],fc80[a,b,c,d],fc81[a,b,c,d],fc82[a,b,c,d],fc83[a,b,c,d],fc84[a,b,c,d]} fd1[a_,b_,c_,d_]:=0 fd2[a_,b_,c_,d_]:=0 fd3[a_,b_,c_,d_]:=0 fd4[a_,b_,c_,d_]:=a^5 fd5[a_,b_,c_,d_]:=0 fd6[a_,b_,c_,d_]:=0 fd7[a_,b_,c_,d_]:=a^4b fd8[a_,b_,c_,d_]:=0 fd9[a_,b_,c_,d_]:=a^4c fd10[a_,b_,c_,d_]:=2a^4d fd11[a_,b_,c_,d_]:=0 fd12[a_,b_,c_,d_]:=0 fd13[a_,b_,c_,d_]:=a^3b^2 fd14[a_,b_,c_,d_]:=0 fd15[a_,b_,c_,d_]:=a^3b c fd16[a_,b_,c_,d_]:=2a^3b d fd17[a_,b_,c_,d_]:=0 fd18[a_,b_,c_,d_]:=a^3c^2 fd19[a_,b_,c_,d_]:=2a^3c d fd20[a_,b_,c_,d_]:=3a^3d^2 fd21[a_,b_,c_,d_]:=0 fd22[a_,b_,c_,d_]:=0 fd23[a_,b_,c_,d_]:=a^2b^3 fd24[a_,b_,c_,d_]:=0 fd25[a_,b_,c_,d_]:=a^2b^2c fd26[a_,b_,c_,d_]:=2a^2b^2d fd27[a_,b_,c_,d_]:=0 fd28[a_,b_,c_,d_]:=a^2b c^2 fd29[a_,b_,c_,d_]:=2a^2b c d fd30[a_,b_,c_,d_]:=3a^2b d^2 fd31[a_,b_,c_,d_]:=0 fd32[a_,b_,c_,d_]:=a^2c^3 fd33[a_,b_,c_,d_]:=2a^2c^2d fd34[a_,b_,c_,d_]:=3a^2c d^2 fd35[a_,b_,c_,d_]:=4a^2d^3 fd36[a_,b_,c_,d_]:=0 fd37[a_,b_,c_,d_]:=0 fd38[a_,b_,c_,d_]:=a b^4 fd39[a_,b_,c_,d_]:=0 fd40[a_,b_,c_,d_]:=a b^3c fd41[a_,b_,c_,d_]:=2a b^3d fd42[a_,b_,c_,d_]:=0 fd43[a_,b_,c_,d_]:=a b^2c^2 fd44[a_,b_,c_,d_]:=2a b^2c d fd45[a_,b_,c_,d_]:=3a b^2d^2 fd46[a_,b_,c_,d_]:=0 fd47[a_,b_,c_,d_]:=a b c^3 fd48[a_,b_,c_,d_]:=2a b c^2d fd49[a_,b_,c_,d_]:=3a b c d^2 fd50[a_,b_,c_,d_]:=4a b d^3 fd51[a_,b_,c_,d_]:=0 fd52[a_,b_,c_,d_]:=a c^4 fd53[a_,b_,c_,d_]:=2a c^3d fd54[a_,b_,c_,d_]:=3a c^2d^2 fd55[a_,b_,c_,d_]:=4a c d^3 fd56[a_,b_,c_,d_]:=5a d^4 fd57[a_,b_,c_,d_]:=0 fd58[a_,b_,c_,d_]:=0 fd59[a_,b_,c_,d_]:=b^5 fd60[a_,b_,c_,d_]:=0 fd61[a_,b_,c_,d_]:=b^4c fd62[a_,b_,c_,d_]:=2b^4d fd63[a_,b_,c_,d_]:=0 fd64[a_,b_,c_,d_]:=b^3c^2 fd65[a_,b_,c_,d_]:=2b^3c d fd66[a_,b_,c_,d_]:=3b^3d^2 fd67[a_,b_,c_,d_]:=0 fd68[a_,b_,c_,d_]:=b^2c^3 fd69[a_,b_,c_,d_]:=2b^2c^2d fd70[a_,b_,c_,d_]:=3b^2c d^2 fd71[a_,b_,c_,d_]:=4b^2d^3 fd72[a_,b_,c_,d_]:=0 fd73[a_,b_,c_,d_]:=b c^4 fd74[a_,b_,c_,d_]:=2b c^3d fd75[a_,b_,c_,d_]:=3b c^2d^2 fd76[a_,b_,c_,d_]:=4b c d^3 fd77[a_,b_,c_,d_]:=5b d^4 fd78[a_,b_,c_,d_]:=0 fd79[a_,b_,c_,d_]:=c^5 fd80[a_,b_,c_,d_]:=2c^4d fd81[a_,b_,c_,d_]:=3c^3d^2 fd82[a_,b_,c_,d_]:=4c^2d^3 fd83[a_,b_,c_,d_]:=5c d^4 fd84[a_,b_,c_,d_]:=6d^5 fd[a_,b_,c_,d_]:={fd1[a,b,c,d],fd2[a,b,c,d],fd3[a,b,c,d],fd4[a,b,c,d],fd5[a,b,c,d],fd6[a,b,c,d],fd7[a,b,c,d],fd8[a,b,c,d],fd9[a,b,c,d],fd10[a,b,c,d],fd11[a,b,c,d],fd12[a,b,c,d],fd13[a,b,c,d],fd14[a,b,c,d],fd15[a,b,c,d],fd16[a,b,c,d],fd17[a,b,c,d],fd18[a,b,c,d],fd19[a,b,c,d],fd20[a,b,c,d],fd21[a,b,c,d],fd22[a,b,c,d],fd23[a,b,c,d],fd24[a,b,c,d],fd25[a,b,c,d],fd26[a,b,c,d],fd27[a,b,c,d],fd28[a,b,c,d],fd29[a,b,c,d],fd30[a,b,c,d],fd31[a,b,c,d],fd32[a,b,c,d],fd33[a,b,c,d],fd34[a,b,c,d],fd35[a,b,c,d],fd36[a,b,c,d],fd37[a,b,c,d],fd38[a,b,c,d],fd39[a,b,c,d],fd40[a,b,c,d],fd41[a,b,c,d],fd42[a,b,c,d],fd43[a,b,c,d],fd44[a,b,c,d],fd45[a,b,c,d],fd46[a,b,c,d],fd47[a,b,c,d],fd48[a,b,c,d],fd49[a,b,c,d],fd50[a,b,c,d],fd51[a,b,c,d],fd52[a,b,c,d],fd53[a,b,c,d],fd54[a,b,c,d],fd55[a,b,c,d],fd56[a,b,c,d],fd57[a,b,c,d],fd58[a,b,c,d],fd59[a,b,c,d],fd60[a,b,c,d],fd61[a,b,c,d],fd62[a,b,c,d],fd63[a,b,c,d],fd64[a,b,c,d],fd65[a,b,c,d],fd66[a,b,c,d],fd67[a,b,c,d],fd68[a,b,c,d],fd69[a,b,c,d],fd70[a,b,c,d],fd71[a,b,c,d],fd72[a,b,c,d],fd73[a,b,c,d],fd74[a,b,c,d],fd75[a,b,c,d],fd76[a,b,c,d],fd77[a,b,c,d],fd78[a,b,c,d],fd79[a,b,c,d],fd80[a,b,c,d],fd81[a,b,c,d],fd82[a,b,c,d],fd83[a,b,c,d],fd84[a,b,c,d]} A46 := { f[0,1,1,1], fa[0,1,1,1], fb[0,1,1,1], fc[0,1,1,1], fd[0,1,1,1], f[0,-1,1,1], fa[0,-1,1,1], fb[0,-1,1,1], fc[0,-1,1,1], fd[0,-1,1,1], f[0,1,-1,1], fa[0,1,-1,1], fb[0,1,-1,1], fc[0,1,-1,1], fd[0,1,-1,1], f[0,1,1,-1], fa[0,1,1,-1], fb[0,1,1,-1], fc[0,1,1,-1], fd[0,1,1,-1], f[1,0,1,1], fa[1,0,1,1], fb[1,0,1,1], fc[1,0,1,1], fd[1,0,1,1], f[-1,0,1,1], fa[-1,0,1,1], fb[-1,0,1,1], fc[-1,0,1,1], fd[-1,0,1,1], f[1,0,-1,1], fa[1,0,-1,1], fb[1,0,-1,1], fc[1,0,-1,1], fd[1,0,-1,1], f[1,0,1,-1], fa[1,0,1,-1], fb[1,0,1,-1], fc[1,0,1,-1], fd[1,0,1,-1], f[1,1,0,1], fa[1,1,0,1], fb[1,1,0,1], fc[1,1,0,1], fd[1,1,0,1], f[-1,1,0,1], fa[-1,1,0,1], fb[-1,1,0,1], fc[-1,1,0,1], fd[-1,1,0,1], f[1,-1,0,1], fa[1,-1,0,1], fb[1,-1,0,1], fc[1,-1,0,1], fd[1,-1,0,1], f[1,1,0,-1], fa[1,1,0,-1], fb[1,1,0,-1], fc[1,1,0,-1], fd[1,1,0,-1], f[1,1,1,0], fa[1,1,1,0], fb[1,1,1,0], fc[1,1,1,0], fd[1,1,1,0], f[-1,1,1,0], fa[-1,1,1,0], fb[-1,1,1,0], fc[-1,1,1,0], fd[-1,1,1,0], f[1,-1,1,0], fa[1,-1,1,0], fb[1,-1,1,0], fc[1,-1,1,0], fd[1,-1,1,0], f[1,1,-1,0], fa[1,1,-1,0], fb[1,1,-1,0], fc[1,1,-1,0], fd[1,1,-1,0], f[1,1,0,0], fa[1,1,0,0], fb[1,1,0,0], fc[1,1,0,0], fd[1,1,0,0], f[-1,1,0,0], fa[-1,1,0,0], fb[-1,1,0,0], fc[-1,1,0,0], fd[-1,1,0,0], f[1,0,1,0], fa[1,0,1,0], fb[1,0,1,0], fc[1,0,1,0], fd[1,0,1,0], f[-1,0,1,0], fa[-1,0,1,0], fb[-1,0,1,0], fc[-1,0,1,0], fd[-1,0,1,0], f[1,0,0,1], fa[1,0,0,1], fb[1,0,0,1], fc[1,0,0,1], fd[1,0,0,1], f[-1,0,0,1], fa[-1,0,0,1], fb[-1,0,0,1], fc[-1,0,0,1], fd[-1,0,0,1], f[0,1,1,0], fa[0,1,1,0], fb[0,1,1,0], fc[0,1,1,0], fd[0,1,1,0], f[0,-1,1,0], fa[0,-1,1,0], fb[0,-1,1,0], fc[0,-1,1,0], fd[0,-1,1,0], f[0,1,0,1], fa[0,1,0,1], fb[0,1,0,1], fc[0,1,0,1], fd[0,1,0,1], f[0,-1,0,1], fa[0,-1,0,1], fb[0,-1,0,1], fc[0,-1,0,1], fd[0,-1,0,1], f[0,0,1,1], fa[0,0,1,1], fb[0,0,1,1], fc[0,0,1,1], fd[0,0,1,1], f[0,0,-1,1], fa[0,0,-1,1], fb[0,0,-1,1], fc[0,0,-1,1], fd[0,0,-1,1]} MatrixRank[A46] (* = 83 *) NullSpace[A46] (* Obtained results must a multiple of g43s[a^2,b^2,c^2,d^2]. *) (*--------------------------- Theorem 2.10 (2) ------------------------------*) h1[a_,b_,c_,d_]:=a^3 h2[a_,b_,c_,d_]:=a^2b h3[a_,b_,c_,d_]:=a^2c h4[a_,b_,c_,d_]:=a^2d h5[a_,b_,c_,d_]:=a b^2 h6[a_,b_,c_,d_]:=a b c h7[a_,b_,c_,d_]:=a b d h8[a_,b_,c_,d_]:=a c^2 h9[a_,b_,c_,d_]:=a c d h10[a_,b_,c_,d_]:=a d^2 h11[a_,b_,c_,d_]:=b^3 h12[a_,b_,c_,d_]:=b^2c h13[a_,b_,c_,d_]:=b^2d h14[a_,b_,c_,d_]:=b c^2 h15[a_,b_,c_,d_]:=b c d h16[a_,b_,c_,d_]:=b d^2 h17[a_,b_,c_,d_]:=c^3 h18[a_,b_,c_,d_]:=c^2d h19[a_,b_,c_,d_]:=c d^2 h20[a_,b_,c_,d_]:=d^3 h[a_,b_,c_,d_]:={h1[a,b,c,d],h2[a,b,c,d],h3[a,b,c,d],h4[a,b,c,d],h5[a,b,c,d],h6[a,b,c,d],h7[a,b,c,d],h8[a,b,c,d],h9[a,b,c,d],h10[a,b,c,d],h11[a,b,c,d],h12[a,b,c,d],h13[a,b,c,d],h14[a,b,c,d],h15[a,b,c,d],h16[a,b,c,d],h17[a,b,c,d],h18[a,b,c,d],h19[a,b,c,d],h20[a,b,c,d]} ha1[a_,b_,c_,d_]:=3a^2 ha2[a_,b_,c_,d_]:=2a b ha3[a_,b_,c_,d_]:=2a c ha4[a_,b_,c_,d_]:=2a d ha5[a_,b_,c_,d_]:=b^2 ha6[a_,b_,c_,d_]:=b c ha7[a_,b_,c_,d_]:=b d ha8[a_,b_,c_,d_]:=c^2 ha9[a_,b_,c_,d_]:=c d ha10[a_,b_,c_,d_]:=d^2 ha11[a_,b_,c_,d_]:=0 ha12[a_,b_,c_,d_]:=0 ha13[a_,b_,c_,d_]:=0 ha14[a_,b_,c_,d_]:=0 ha15[a_,b_,c_,d_]:=0 ha16[a_,b_,c_,d_]:=0 ha17[a_,b_,c_,d_]:=0 ha18[a_,b_,c_,d_]:=0 ha19[a_,b_,c_,d_]:=0 ha20[a_,b_,c_,d_]:=0 ha[a_,b_,c_,d_]:={ha1[a,b,c,d],ha2[a,b,c,d],ha3[a,b,c,d],ha4[a,b,c,d],ha5[a,b,c,d],ha6[a,b,c,d],ha7[a,b,c,d],ha8[a,b,c,d],ha9[a,b,c,d],ha10[a,b,c,d],ha11[a,b,c,d],ha12[a,b,c,d],ha13[a,b,c,d],ha14[a,b,c,d],ha15[a,b,c,d],ha16[a,b,c,d],ha17[a,b,c,d],ha18[a,b,c,d],ha19[a,b,c,d],ha20[a,b,c,d]} hb1[a_,b_,c_,d_]:=0 hb2[a_,b_,c_,d_]:=a^2 hb3[a_,b_,c_,d_]:=0 hb4[a_,b_,c_,d_]:=0 hb5[a_,b_,c_,d_]:=2a b hb6[a_,b_,c_,d_]:=a c hb7[a_,b_,c_,d_]:=a d hb8[a_,b_,c_,d_]:=0 hb9[a_,b_,c_,d_]:=0 hb10[a_,b_,c_,d_]:=0 hb11[a_,b_,c_,d_]:=3b^2 hb12[a_,b_,c_,d_]:=2b c hb13[a_,b_,c_,d_]:=2b d hb14[a_,b_,c_,d_]:=c^2 hb15[a_,b_,c_,d_]:=c d hb16[a_,b_,c_,d_]:=d^2 hb17[a_,b_,c_,d_]:=0 hb18[a_,b_,c_,d_]:=0 hb19[a_,b_,c_,d_]:=0 hb20[a_,b_,c_,d_]:=0 hb[a_,b_,c_,d_]:={hb1[a,b,c,d],hb2[a,b,c,d],hb3[a,b,c,d],hb4[a,b,c,d],hb5[a,b,c,d],hb6[a,b,c,d],hb7[a,b,c,d],hb8[a,b,c,d],hb9[a,b,c,d],hb10[a,b,c,d],hb11[a,b,c,d],hb12[a,b,c,d],hb13[a,b,c,d],hb14[a,b,c,d],hb15[a,b,c,d],hb16[a,b,c,d],hb17[a,b,c,d],hb18[a,b,c,d],hb19[a,b,c,d],hb20[a,b,c,d]} hc1[a_,b_,c_,d_]:=0 hc2[a_,b_,c_,d_]:=0 hc3[a_,b_,c_,d_]:=a^2 hc4[a_,b_,c_,d_]:=0 hc5[a_,b_,c_,d_]:=0 hc6[a_,b_,c_,d_]:=a b hc7[a_,b_,c_,d_]:=0 hc8[a_,b_,c_,d_]:=2a c hc9[a_,b_,c_,d_]:=a d hc10[a_,b_,c_,d_]:=0 hc11[a_,b_,c_,d_]:=0 hc12[a_,b_,c_,d_]:=b^2 hc13[a_,b_,c_,d_]:=0 hc14[a_,b_,c_,d_]:=2b c hc15[a_,b_,c_,d_]:=b d hc16[a_,b_,c_,d_]:=0 hc17[a_,b_,c_,d_]:=3c^2 hc18[a_,b_,c_,d_]:=2c d hc19[a_,b_,c_,d_]:=d^2 hc20[a_,b_,c_,d_]:=0 hc[a_,b_,c_,d_]:={hc1[a,b,c,d],hc2[a,b,c,d],hc3[a,b,c,d],hc4[a,b,c,d],hc5[a,b,c,d],hc6[a,b,c,d],hc7[a,b,c,d],hc8[a,b,c,d],hc9[a,b,c,d],hc10[a,b,c,d],hc11[a,b,c,d],hc12[a,b,c,d],hc13[a,b,c,d],hc14[a,b,c,d],hc15[a,b,c,d],hc16[a,b,c,d],hc17[a,b,c,d],hc18[a,b,c,d],hc19[a,b,c,d],hc20[a,b,c,d]} hd1[a_,b_,c_,d_]:=0 hd2[a_,b_,c_,d_]:=0 hd3[a_,b_,c_,d_]:=0 hd4[a_,b_,c_,d_]:=a^2 hd5[a_,b_,c_,d_]:=0 hd6[a_,b_,c_,d_]:=0 hd7[a_,b_,c_,d_]:=a b hd8[a_,b_,c_,d_]:=0 hd9[a_,b_,c_,d_]:=a c hd10[a_,b_,c_,d_]:=2a d hd11[a_,b_,c_,d_]:=0 hd12[a_,b_,c_,d_]:=0 hd13[a_,b_,c_,d_]:=b^2 hd14[a_,b_,c_,d_]:=0 hd15[a_,b_,c_,d_]:=b c hd16[a_,b_,c_,d_]:=2b d hd17[a_,b_,c_,d_]:=0 hd18[a_,b_,c_,d_]:=c^2 hd19[a_,b_,c_,d_]:=2c d hd20[a_,b_,c_,d_]:=3d^2 hd[a_,b_,c_,d_]:={hd1[a,b,c,d],hd2[a,b,c,d],hd3[a,b,c,d],hd4[a,b,c,d],hd5[a,b,c,d],hd6[a,b,c,d],hd7[a,b,c,d],hd8[a,b,c,d],hd9[a,b,c,d],hd10[a,b,c,d],hd11[a,b,c,d],hd12[a,b,c,d],hd13[a,b,c,d],hd14[a,b,c,d],hd15[a,b,c,d],hd16[a,b,c,d],hd17[a,b,c,d],hd18[a,b,c,d],hd19[a,b,c,d],hd20[a,b,c,d]} A43 := { h[0,1,1,1], hb[0,1,1,1], hc[0,1,1,1], h[1,0,1,1], ha[1,0,1,1], hc[1,0,1,1], h[1,1,0,1], ha[1,1,0,1], hb[1,1,0,1], h[1,1,1,0], ha[1,1,1,0], hb[1,1,1,0], h[1,1,0,0], ha[1,1,0,0], h[1,0,1,0], ha[1,0,1,0], h[1,0,0,1], ha[1,0,0,1], h[0,1,1,0], hb[0,1,1,0], h[0,1,0,1], hb[0,1,0,1], h[0,0,1,1], hc[0,0,1,1]} MatrixRank[A43] (* = 19 *) NullSpace[A43] (* Obtained results must a multiple of g43s[a,b,c,d]. *) B43 := { h[0,1,1,1], h[0,-1,1,1], h[0,1,-1,1], h[0,1,1,-1], h[1,0,1,1], h[-1,0,1,1], h[1,0,-1,1], h[1,0,1,-1], h[1,1,0,1], h[-1,1,0,1], h[1,-1,0,1], h[1,1,0,-1], h[1,1,1,0], h[-1,1,1,0], h[1,-1,1,0], h[1,1,-1,0], h[1,1,0,0], h[-1,1,0,0], h[1,0,1,0], h[-1,0,1,0], h[1,0,0,1], h[-1,0,0,1], h[0,1,1,0], h[0,-1,1,0], h[0,1,0,1], h[0,-1,0,1], h[0,0,1,1], h[0,0,-1,1]} MatrixRank[B43] (* = 20 * NullSpace[B43] (* {} Null *) (*===========================================================================*) (*============================== Section 3 ==================================*) (*===========================================================================*) S6[a_,b_,c_]:=(a^6+b^6+c^6) S51[a_,b_,c_]:=(a^5b + b^5c + c^5a) S15[a_,b_,c_]:=(a b^5 + b c^5 + c a^5) S42[a_,b_,c_]:=(a^4b^2 + b^4c^2 + c^4a^2) S24[a_,b_,c_]:=(a^2b^4 + b^2c^4 + c^2a^4) S33[a_,b_,c_]:=(a^3b^3 + b^3c^3 + c^3a^3) US3[a_,b_,c_]:=a b c(a^3 + b^3 + c^3) US21[a_,b_,c_]:=a b c(a^2b + b^2c + c^2a) US12[a_,b_,c_]:=a b c(a b^2 + b c^2+ c a^2) U2[a_,b_,c_]:=(a^2b^2c^2) T51[a_,b_,c_]:=S51[a,b,c]+S15[a,b,c] T42[a_,b_,c_]:=S42[a,b,c]+S24[a,b,c] UT21[a_,b_,c_]:=US21[a,b,c]+US12[a,b,c] S5[a_, b_, c_] := a^5 + b^5 + c^5 S41[a_, b_, c_] := a^4 b + b^4 c + c^4 a S14[a_, b_, c_] := a b^4 + b c^4 + c a^4 S32[a_, b_, c_] := a^3 b^2 + b^3 c^2 + c^3 a^2 S23[a_, b_, c_] := a^2 b^3 + b^2 c^3 + c^2 a^3 T41[a_, b_, c_] := S41[a,b,c] + S14[a,b,c] T32[a_, b_, c_] := S32[a,b,c] + S23[a,b,c] US2[a_, b_, c_] := a b c(a^2+b^2+c^2) US11[a_, b_, c_] := a b c(a b+b c+c a) S4[a_,b_,c_] := (a^4+b^4+c^4) S31[a_,b_,c_] := (a^3b + b^3c + c^3a) S13[a_,b_,c_] := (a b^3 + b c^3 + c a^3) S22[a_,b_,c_] := (a^2b^2 + b^2c^2 + c^2a^2) US1[a_,b_,c_] := a b c(a + b + c) T31[a_,b_,c_] := (a^3b + b^3c + c^3a)+(a b^3 + b c^3 + c a^3) S3[a_,b_,c_] := (a^3+b^3+c^3) S21[a_,b_,c_] := (a^2b + b^2c + c^2a) S12[a_,b_,c_] := (a b^2 + b c^2 + c a^2) U[a_,b_,c_] := a b c T21[a_,b_,c_] := (a^2b + b^2c + c^2a)+(a b^2 + b c^2 + c a^2) S2[a_,b_,c_] := (a^2+b^2+c^2) S11[a_,b_,c_] := (a b + b c + c a) S1[a_, b_, c_] := a + b + c S120[a_,b_,c_] := a^(12)+b^(12)+c^(12) T111[a_,b_,c_] := a^(11)b+b^(11)c+c^(11)a+a b^(11)+b c^(11)+c a^(11) T102[a_,b_,c_] := a^(10)b^2+b^(10)c^2+c^(10)a^2+a^2b^(10)+b^2c^(10)+c^2a^(10) T93[a_,b_,c_] := a^9b^3+b^9c^3+c^9a^3+a^3b^9+b^3c^9+c^3a^9 T84[a_,b_,c_] := a^8b^4+b^8c^4+c^8a^4+a^4b^8+b^4c^8+c^4a^8 T75[a_,b_,c_] := a^7b^5+b^7c^5+c^7a^5+a^5b^7+b^5c^7+c^5a^7 S66[a_,b_,c_] := a^6b^6+b^6c^6+c^6a^6 S1011[a_,b_,c_] := a b c(a^9+b^9+c^9) T921[a_,b_,c_] := a b c(a^8b+b^8c+c^8a+a b^8+b c^8+c a^8) T831[a_,b_,c_] := a b c(a^7b^2+b^7c^2+c^7a^2+a^2b^7+b^2c^7+c^2a^7) T741[a_,b_,c_] := a b c(a^6b^3+b^6c^3+c^6a^3+a^3b^6+b^3c^6+c^3a^6) T651[a_,b_,c_] := a b c(a^5b^4+b^5c^4+c^5a^4+a^4b^5+b^4c^5+c^4a^5) S822[a_,b_,c_] := a^2b^2c^2(a^6+b^6+c^6) T732[a_,b_,c_] := a^2b^2c^2(a^5b+b^5c+c^5a+a b^5+b c^5+c a^5) T642[a_,b_,c_] := a^2b^2c^2(a^4b^2+b^4c^2+c^4a^2+a^2b^4+b^2c^4+c^2a^4) S552[a_,b_,c_] := a^2b^2c^2(a^3b^3+b^3c^3+c^3a^3) S633[a_,b_,c_] := a^3b^3c^3(a^3+b^3+c^3) T543[a_,b_,c_] := a^3b^3c^3(a^2b+b^2c+c^2a+a b^2+b c^2+c a^2) U4[a_,b_,c_] := a^4b^4c^4 S10[a_,b_,c_] := a^(10)+b^(10)+c^(10) T91[a_,b_,c_] := a^9b+b^9c+c^9a+a b^9+b c^9+c a^9 T82[a_,b_,c_] := a^8b^2+b^8c^2+c^8a^2+a^2b^8+b^2c^8+c^2a^8 T73[a_,b_,c_] := a^7b^3+b^7c^3+c^7a^3+a^3b^7+b^3c^7+c^3a^7 T64[a_,b_,c_] := a^6b^4+b^6c^4+c^6a^4+a^4b^6+b^4c^6+c^4a^6 S55[a_,b_,c_] := a^5b^5+b^5c^5+c^5a^5 S811[a_,b_,c_] := a b c(a^7+b^7+c^7) T721[a_,b_,c_] := a b c(a^6b+b^6c+c^6a+a b^6+b c^6+c a^6) T631[a_,b_,c_] := a b c(a^5b^2+b^5c^2+c^5a^2+a^2b^5+b^2c^5+c^2a^5) T541[a_,b_,c_] := a b c(a^4b^3+b^4c^3+c^4a^3+a^3b^4+b^3c^4+c^3a^4) S622[a_,b_,c_] := a^2b^2c^2(a^4+b^4+c^4) T532[a_,b_,c_] := a^2b^2c^2(a^3b+b^3c+c^3a+a b^3+b c^3+c a^3) S442[a_,b_,c_] := a^2b^2c^2(a^2b^2+b^2c^2+c^2a^2) S433[a_,b_,c_] := a^3b^3c^3(a+b+c) S8[a_,b_,c_] := a^8+b^8+c^8 T71[a_,b_,c_] := a^7b+b^7c+c^7a+a b^7+b c^7+c a^7 T62[a_,b_,c_] := a^6b^2+b^6c^2+c^6a^2+a^2b^6+b^2c^6+c^2a^6 T53[a_,b_,c_] := a^5b^3+b^5c^3+c^5a^3+a^3b^5+b^3c^5+c^3a^5 S44[a_,b_,c_] := a^4b^4+b^4c^4+c^4a^4 S611[a_,b_,c_] := a b c(a^5+b^5+c^5) T521[a_,b_,c_] := a b c(a^4b+b^4c+c^4a+a b^4+b c^4+c a^4) T431[a_,b_,c_] := a b c(a^3b^2+b^3c^2+c^3a^2+a^2b^3+b^2c^3+c^2a^3) S422[a_,b_,c_] := a^2b^2c^2(a^2+b^2+c^2) S332[a_,b_,c_] := a^2b^2c^2(a b+b c+c a) (*---------------------------- Proposition 3.4 ------------------------------*) Delta1[a_,b_,c_,w_] := (2 S3[a,b,c] - (w+2) T21[a,b,c] + (6w+6) U[a,b,c]) Delta2[a_,b_,c_,w_] := ((2w+1)S2[a,b,c] - (w^2+2)S11[a,b,c]) Xi[a_,b_,c_,w_] := (a+b+c) (1-w) Delta1[a,b,c,w] Delta2[a,b,c,w] cf1[x_,y_,z_]:=x^3 cf2[x_,y_,z_]:=x^2y cf3[x_,y_,z_]:=x^2z cf4[x_,y_,z_]:=x y^2 cf5[x_,y_,z_]:=x y z cf6[x_,y_,z_]:=x z^2 cf7[x_,y_,z_]:=y^3 cf8[x_,y_,z_]:=y^2z cf9[x_,y_,z_]:=y z^2 cf10[x_,y_,z_]:=z^3 cf[x_,y_,z_] := {cf1[x,y,z],cf2[x,y,z],cf3[x,y,z],cf4[x,y,z],cf5[x,y,z],cf6[x,y,z],cf7[x,y,z],cf8[x,y,z],cf9[x,y,z],cf10[x,y,z]} A[a_,b_,c_,w_] :={cf[1,1,1], cf[a,b,c],cf[a,c,b],cf[b,a,c],cf[b,c,a],cf[c,a,b],cf[c,b,a], cf[w,1,1],cf[1,w,1],cf[1,1,w]} (* The following takes long time. *) Factor[Det[A[a,b,c,w]] - (a-b)^3(b-c)^3(c-a)^3(a+b+c)^2 (w-1)^4 Delta1[a,b,c,w] Delta2[a,b,c,w]^2] (* = 0 *) (*------------------------- Definition of FrakF[] ---------------------------*) p0F[a_,b_,c_,w_] := T102[a,b,c] + (4 w^2-w-4) T93[a,b,c] + (-4 w^3-w^2+2 w+4) T84[a,b,c] + (w^4-4 w^2+w+4) T75[a,b,c] + (2 w^4+8 w^3+2 w^2-4 w-10) S66[a,b,c] + 2 S1011[a,b,c] + (-4 w^2-7 w-8) T921[a,b,c] + (w^2+16 w+12) T831[a,b,c] + (w^4+8 w^3-9 w^2-34 w-8) T741[a,b,c] + (-13 w^4-16 w^3+12 w^2+25 w+2) T651[a,b,c] + (8 w^3+24 w^2+60 w+28) S822[a,b,c] + (-2 w^4-16 w^3-59 w^2-87 w-36) T732[a,b,c] + (26 w^4+84 w^3+135 w^2+88 w+23) T642[a,b,c] + (12 w^4-56 w^3-168 w^2-108 w-16) S552[a,b,c] + (-26 w^4-56 w^3+62 w^2+102 w+30) S633[a,b,c] + (-12 w^4-48 w^3-80 w^2-38 w-2) T543[a,b,c] + (30 w^4+240 w^3+270 w^2+60 w-30) U4[a,b,c] p1F[a_,b_,c_,w_] := -2 T111[a,b,c] + (-4 w^2+2 w+6) T102[a,b,c] + (-2 w^2-2 w-2) T93[a,b,c] + (3 w^4+4 w^3+5 w^2-4 w-8) T84[a,b,c] + (-w^5-w^4+2 w^2+2 w+4) T75[a,b,c] + (-2 w^5-8 w^4-8 w^3-2 w^2+4 w+4) S66[a,b,c] + (8 w^2+8 w+8) S1011[a,b,c] + (-2 w^2-26 w-16) T921[a,b,c] + (-2 w^3+13 w^2+49 w+18) T831[a,b,c] + (-w^5-7 w^4-10 w^3-17 w^2-11 w-8) T741[a,b,c] + (13 w^5+25 w^4+20 w^3-2 w^2-20 w) T651[a,b,c] + (-6 w^4-12 w^3-12 w^2-6 w) S822[a,b,c] + (2 w^5+14 w^4+42 w^3+27 w^2+9 w+14) T732[a,b,c] + (-26 w^5-89 w^4-136 w^3-87 w^2-24 w-10) T642[a,b,c] + (-12 w^5+30 w^4+108 w^3+132 w^2+90 w+12) S552[a,b,c] + (26 w^5+68 w^4+32 w^3-18 w^2-60 w-24) S633[a,b,c] + (12 w^5+48 w^4+72 w^3+48 w^2+12 w-6) T543[a,b,c] + (-30 w^5-210 w^4-300 w^3-210 w^2-30 w+60) U4[a,b,c] p2F[a_,b_,c_,w_] := S120[a,b,c] + (-4 w^2-w) T111[a,b,c] + (12 w^3+7 w^2-2 w-7) T102[a,b,c] + (-9 w^4-8 w^3-2 w^2+4 w+4) T93[a,b,c] + (2 w^5-4 w^4-12 w^3-3 w^2+6 w+11) T84[a,b,c] + (4 w^5+4 w^4+8 w^3+6 w^2-3 w-4) T75[a,b,c] + (4 w^5-2 w^4-8 w^2-8 w-10) S66[a,b,c] + (8 w^3+5 w^2+8 w) S1011[a,b,c] + (-15 w^4-30 w^3-12 w^2+18 w+10) T921[a,b,c] + (8 w^5+47 w^4+54 w^3-4 w^2-52 w-20) T831[a,b,c] + (-w^6-13 w^5-4 w^4+10 w^3+4 w^2-5 w) T741[a,b,c] + (-3 w^6-15 w^5-24 w^4-2 w^3+11 w^2+32 w+10) T651[a,b,c] + (12 w^5+33 w^4-12 w^3-24 w^2-36 w-9) S822[a,b,c] + (-3 w^6-32 w^5-44 w^4-22 w^3+100 w^2+79 w+30) T732[a,b,c] + (12 w^6+36 w^5+42 w^4-68 w^3-158 w^2-110 w-36) T642[a,b,c] + (3 w^6+12 w^5+36 w^4+48 w^3+54 w^2+102 w+24) S552[a,b,c] + (3 w^6+2 w^5-10 w^4-20 w^3+33 w^2-24 w-20) S633[a,b,c] + (-6 w^6-3 w^5-6 w^4+60 w^3+57 w^2+33 w+6) T543[a,b,c] + (-12 w^6-12 w^5-93 w^4-84 w^3-192 w^2-120 w+18) U4[a,b,c] p3F[a_,b_,c_,w_] := (4 w^2-2) S120[a,b,c] + (-8 w^3-4 w^2+2 w+4) T111[a,b,c] + (5 w^4+4 w^3+5 w^2) T102[a,b,c] + (-w^5+3 w^4-8 w^3-8 w^2-2 w+4) T93[a,b,c] + (-2 w^5+12 w^4+8 w^3+6 w^2-8 w-14) T84[a,b,c] + (-5 w^5+7 w^4+16 w^3+12 w^2-8) T75[a,b,c] + (-8 w^5 - 14 w^4 - 24 w^3 - 30 w^2 + 16 w + 32) S66[a,b,c] + (14 w^4+24 w^3+12 w^2-32 w-20) S1011[a,b,c] + (-7 w^5-27 w^4-36 w^3+4 w^2+30 w+28) T921[a,b,c] + (w^6+7 w^4+16 w^3-10 w^2-26 w-20) T831[a,b,c] + (3 w^6-6 w^5-49 w^4-80 w^3+6 w^2+100 w+32) T741[a,b,c] + (4 w^6+29 w^5+35 w^4+4 w^3-8 w^2-74 w-24) T651[a,b,c] + (2 w^6+36 w^5+84 w^4+144 w^3+48 w^2-36 w-38) S822[a,b,c] + (-7 w^6-27 w^5-122 w^4-96 w^3-130 w^2-2 w-16) T732[a,b,c] + (2 w^6+68 w^5+175 w^4+292 w^3+217 w^2+92 w+46) T642[a,b,c] + (-32 w^6-108 w^5-222 w^4-216 w^3-48 w^2-168 w-40) S552[a,b,c] + (6 w^6+46 w^5+68 w^4+112 w^3-120 w^2-36 w+28) S633[a,b,c] + (-56 w^5-92 w^4-200 w^3-92 w^2-14 w+4) T543[a,b,c] + (54 w^6+144 w^5+486 w^4+408 w^3+414 w^2+180 w-96) U4[a,b,c] p4F[a_,b_,c_,w_] := 2 S120[a,b,c] + (8 w^2+2 w-4) T111[a,b,c] + (-8 w^3-4 w^2-4 w-2) T102[a,b,c] + (6 w^4+10 w^3+9 w^2+3 w+4) T93[a,b,c] + (-4 w^5-w^4-5 w^2-2 w+6) T84[a,b,c] + (w^6-4 w^5+5 w^4-10 w^3-17 w^2-5 w) T75[a,b,c] + (2 w^6+24 w^4+16 w^3+18 w^2+12 w-12) S66[a,b,c] + (-48 w^3-66 w^2-48 w) S1011[a,b,c] + (42 w^4+98 w^3+143 w^2+89 w+36) T921[a,b,c] + (-16 w^5-88 w^4-188 w^3-242 w^2-140 w-58) T831[a,b,c] + (3 w^6+30 w^5+69 w^4+168 w^3+222 w^2+156 w+18) T741[a,b,c] + (-7 w^6-22 w^5-67 w^4-118 w^3-65 w^2-59 w+8) T651[a,b,c] + (-24 w^5-96 w^4-120 w^3-138 w^2-84 w-42) S822[a,b,c] + (4 w^6+56 w^5+98 w^4+226 w^3+103 w^2+67 w-2) T732[a,b,c] + (2 w^6+32 w^5+23 w^4+60 w^3+67 w^2+52 w+40) T642[a,b,c] + (6 w^6+24 w^5+42 w^4-78 w^2-240 w-60) S552[a,b,c] + (-32 w^6-108 w^5-252 w^4-300 w^3-306 w^2-18 w+80) S633[a,b,c] + (-42 w^5-12 w^4-60 w^3+12 w^2-24) T543[a,b,c] + (54 w^6+144 w^5+396 w^4+288 w^3+324 w^2+180 w-36) U4[a,b,c] p5F[a_,b_,c_,w_] := (-4 w^2-4 w-2) S120[a,b,c] + (8 w^3+10 w^2+10 w+6) T111[a,b,c] + (-9 w^4-22 w^3-24 w^2-9 w-4) T102[a,b,c] + (5 w^5+11 w^4+22 w^3+19 w^2+7 w-2) T93[a,b,c] + (-w^6+w^5-8 w^4+4 w^3+4 w^2-2 w+2) T84[a,b,c] + (-w^6-21 w^4-30 w^3-29 w^2-17 w-4) T75[a,b,c] + (8 w^5+14 w^4+36 w^3+48 w^2+30 w+8) S66[a,b,c] + (-6 w^4+4 w^3-10 w^2+2 w-8) S1011[a,b,c] + (3 w^5+9 w^4+14 w^3-37 w^2-31 w-20) T921[a,b,c] + (-w^6-2 w^5+w^4+46 w^3+129 w^2+81 w+40) T831[a,b,c] + (-2 w^6-7 w^5-19 w^4-78 w^3-119 w^2-107 w-10) T741[a,b,c] + (5 w^6+2 w^5+41 w^4+86 w^3+27 w^2+45 w-8) T651[a,b,c] + (-18 w^5-24 w^4-36 w^3+42 w^2+66 w+42) S822[a,b,c] + (5 w^6+9 w^5+42 w^4-108 w^3-70 w^2-82 w-12) T732[a,b,c] + (-7 w^6-37 w^5-19 w^4-22 w^3-16 w^2-25 w-30) T642[a,b,c] + (6 w^6+6 w^5-42 w^4-60 w^3-30 w^2+162 w+48) S552[a,b,c] + (6 w^6+6 w^5+96 w^4+184 w^3+272 w^2+44 w-56) S633[a,b,c] + (24 w^5-36 w^4+24 w^3+6 w^2+30 w+30) T543[a,b,c] + (-24 w^6+6 w^5-66 w^4-48 w^3-354 w^2-300 w-24) U4[a,b,c] FrakF[a_,b_,c_,u1_,u2_,u3_,w_] := p0F[u1,u2,u3,w] s0[a,b,c] + p1F[u1,u2,u3,w] s1[a,b,c] + p2F[u1,u2,u3,w] s2[a,b,c] + p3F[u1,u2,u3,w] s3[a,b,c] + p4F[u1,u2,u3,w] s4[a,b,c] + p5F[u1,u2,u3,w] s5[a,b,c] (*---------------------------- Proposition 3.5 ------------------------------*) s0[a_,b_,c_] := S6[a,b,c] - 3 U2[a,b,c] s1[a_,b_,c_] := T51[a,b,c] - 6 U2[a,b,c] s2[a_,b_,c_] := T42[a,b,c] - 6 U2[a,b,c] s3[a_,b_,c_] := S33[a,b,c] - 3 U2[a,b,c] s4[a_,b_,c_] := US3[a,b,c] - 3 U2[a,b,c] s5[a_,b_,c_] := UT21[a,b,c] - 6 U2[a,b,c] s6[a_,b_,c_] := U2[a,b,c] s[a_,b_,c_] := {s0[a,b,c], s1[a,b,c], s2[a,b,c], s3[a,b,c], s4[a,b,c], s5[a,b,c]} s0a[a_,b_,c_] := 6 a^5 - 6 a b^2 c^2 s1a[a_,b_,c_] := 5 a^4 b + b^5 + 5 a^4 c - 12 a b^2 c^2 + c^5 s2a[a_,b_,c_] := 4 a^3 b^2 + 2 a b^4 + 4 a^3 c^2 - 12 a b^2 c^2 + 2 a c^4 s3a[a_,b_,c_] := 3 a^2 b^3 - 6 a b^2 c^2 + 3 a^2 c^3 s4a[a_,b_,c_] := 4 a^3 b c + b^4 c - 6 a b^2 c^2 + b c^4 s5a[a_,b_,c_] := 3a^2b^2c+2a b^3c+3a^2b c^2-12a b^2c^2+b^3c^2+2a b c^3+b^2c^3 s6a[a_,b_,c_] := 2a b^2c^2 sa[a_,b_,c_] := {s0a[a,b,c], s1a[a,b,c], s2a[a,b,c], s3a[a,b,c], s4a[a,b,c], s5a[a,b,c]} sb[a_,b_,c_] := sa[b,a,c] sc[a_,b_,c_] := sa[c,a,b] s0aa[a_,b_,c_] := 30 a^4 - 6 b^2 c^2 s1aa[a_,b_,c_] := 20 a^3 b + 20 a^3 c - 12 b^2 c^2 s2aa[a_,b_,c_] := 12 a^2 b^2 + 2 b^4 + 12 a^2 c^2 - 12 b^2 c^2 + 2 c^4 s3aa[a_,b_,c_] := 6 a b^3 - 6 b^2 c^2 + 6 a c^3 s4aa[a_,b_,c_] := 12 a^2 b c - 6 b^2 c^2 s5aa[a_,b_,c_] := 6 a b^2 c + 2 b^3 c + 6 a b c^2 - 12 b^2 c^2 + 2 b c^3 s6aa[a_,b_,c_] := 2b^2 c^2 saa[a_,b_,c_] := {s0aa[a,b,c], s1aa[a,b,c], s2aa[a,b,c], s3aa[a,b,c], s4aa[a,b,c], s5aa[a,b,c]} sbb[a_,b_,c_] := saa[b,a,c] sbc[a_,b_,c_] := saa[c,a,b] s0aaa[a_,b_,c_] := 120 a^3 s1aaa[a_,b_,c_] := 60 a^2 b + 60 a^2 c s2aaa[a_,b_,c_] := 24 a b^2 + 24 a c^2 s3aaa[a_,b_,c_] := 6 b^3 + 6 c^3 s4aaa[a_,b_,c_] := 24 a b c s5aaa[a_,b_,c_] := 6 b^2 c + 6 b c^2 s6aaa[a_,b_,c_] := 0 saaa[a_,b_,c_] := {s0aaa[a,b,c], s1aaa[a,b,c], s2aaa[a,b,c], s3aaa[a,b,c], s4aaa[a,b,c], s5aaa[a,b,c]} A[u_,v_,w_] :={s[u,v,1],sa[u,v,1],sb[u,v,1],s[w,1,1],sa[w,1,1],sb[w,1,1]} Factor[NullSpace[A[s,t,u]]] (* This takes very long time. *) (*----------------------------- Definition 3.7 ------------------------------*) Delta3[a_,b_,c_,w_] := S4[a,b,c]-(w+1)T31[a,b,c]+(w^2+2w)S22[a,b,c]-(w^2-1)US1[a,b,c] Delta4[a_,b_,c_,w_] := 2 S5[a,b,c] - (2w+3) T41[a,b,c] + (-w^2+2w+1) T32[a,b,c] + 4(w+1)^2 US2[a,b,c] - (2w^2+8w+2) US11[a,b,c] Delta5[a_,b_,c_,w_] := ((w+1)a-b-c)((w+1)b-c-a)((w+1)c-a-b) Factor[Delta5[a,b,c,w] - ( (w+1)S3[a,b,c] - (w^2+w+1)T21[a,b,c] + (w^3+3w^2+6w+2) U[a,b,c])] (* = 0 *) Factor[Delta1[x,1,1,w]] (* = 2(x-1)^2(x-w). not PSD *) Factor[Delta2[x,1,1,w]] (* = (x-w)((2w+1)x+(w-4)). not PSD *) Factor[Delta3[x,1,1,w]] (* = (x-1)^2(x-w)^2 >= 0. PSD *) Factor[Delta4[x,1,1,w]] (* = 2(x-1)^3(x-w)^2. not PSD evenif x>=0. *) Factor[Delta5[x,1,1,w]] (* = (x-w)^2((w+1)x-2). not PSD *) Factor[Delta5[x,1,0,w]] (* = (x+1)(x-w-1)((w+1)x-1) *) (*------------------------------- $h_1(s)$ ----------------------------------*) h1s0[a_,b_,c_,w_] := S120[a,b,c] (4 - 8 w - 32 w^2) + T111[a,b,c] (-32+20w+92w^2+64w^3) + T102[a,b,c] (104 + 2 w - 166 w^2 - 180 w^3 - 48 w^4) + T93[a,b,c] (-160 - 36 w + 204 w^2 + 176 w^3 + 88 w^4 + 16 w^5) + T84[a,b,c] (60 + 16 w - 8 w^2 - 16 w^3 - 40 w^4 - 10 w^5 - 2 w^6) + T75[a,b,c] (192 + 16 w - 296 w^2 - 240 w^3 - 88 w^4 - 16 w^5) + S66[a,b,c] (-336 - 20 w + 412 w^2 + 392 w^3 + 176 w^4 + 20 w^5 + 4 w^6) + S1011[a,b,c] (200 + 84 w - 216 w^2 - 296 w^3 - 96 w^4) + T921[a,b,c] (-504 - 532 w + 176 w^2 + 680 w^3 + 384 w^4 + 48 w^5) + T831[a,b,c] (640 + 1132 w - 32 w^2 - 824 w^3 - 592 w^4 - 100 w^5 - 8 w^6) + T741[a,b,c] (-416 - 1344 w - 180 w^2 + 712 w^3 + 408 w^4 + 180 w^5 - 8 w^6) + T651[a,b,c] (112 + 640 w + 160 w^2 - 336 w^3 - 232 w^4 - 16 w^5 - 4 w^6) + S822[a,b,c] (972 + 1872 w + 216 w^2 - 960 w^3 - 1044 w^4 - 396 w^5 - 12 w^6) + T732[a,b,c] (-952 - 2684 w - 1208 w^2 + 856 w^3 + 1400 w^4 + 560 w^5 + 84 w^6) + T642[a,b,c] (524 + 2350 w + 2782 w^2 + 68 w^3 - 1288 w^4 - 862 w^5 - 118 w^6) + S552[a,b,c] (-288 - 2016 w - 3600 w^2 + 96 w^3 + 1512 w^4 + 540 w^5 + 192 w^6) + S633[a,b,c] (616 + 2548 w + 1768 w^2 - 1896 w^3 - 856 w^4 - 340 w^5 - 112 w^6) + T543[a,b,c] (-144 - 960 w - 1500 w^2 - 72 w^3 + 504 w^4 + 396 w^5 + 48 w^6) + U4[a,b,c] (-48 + 900 w + 4212 w^2 + 2664 w^3 - 2052 w^4 - 648 w^5 - 168 w^6) h1s1[a_,b_,c_,w_] := S120[a,b,c] (28 + 52 w - 8 w^2) + T111[a,b,c] (-136 - 144 w - 6 w^2 + 16 w^3) + T102[a,b,c] (254 + 183 w + 75 w^2 - 14 w^3 + 42 w^4) + T93[a,b,c] (-192 - 180 w - 72 w^2 - 20 w^3 - 8 w^4 - 50 w^5) + T84[a,b,c] (-68 + 24 w - 48 w^2 + 64 w^3 + 30 w^4 - 15 w^5 + 13 w^6) + T75[a,b,c] (328 + 324 w + 78 w^2 + 4 w^3 + 72 w^4 - 6 w^5 + 10 w^6) + S66[a,b,c] (-428 - 518 w - 38 w^2 - 100 w^3 - 16 w^4 - 82 w^5 - 6 w^6) + S1011[a,b,c] (468 + 462 w + 78 w^2 - 444 w^3 - 132 w^4) + T921[a,b,c] (-664 - 924 w + 198 w^2 + 724 w^3 + 492 w^4 + 66 w^5) + T831[a,b,c] (628 + 1338 w - 636 w^2 - 1472 w^3 - 888 w^4 - 102 w^5 - 2 w^6) + T741[a,b,c] (-540 - 930 w + 510 w^2 + 1536 w^3 + 840 w^4 + 372 w^5 - 6 w^6) + T651[a,b,c] (244 + 198 w - 144 w^2 - 872 w^3 - 360 w^4 - 38 w^6) + S822[a,b,c] (684 + 1416 w - 96 w^2 - 1032 w^3 - 1416 w^4 - 498 w^5 - 30 w^6) + T732[a,b,c] (-620 - 1842 w - 222 w^2 + 2668 w^3 + 2004 w^4 + 834 w^5 + 94 w^6) + T642[a,b,c] (598 + 2145 w + 1413 w^2 - 2450 w^3 - 3504 w^4 - 1557 w^5 - 209 w^6) + S552[a,b,c] (-504 - 1956 w - 1200 w^2 + 2832 w^3 + 3252 w^4 + 1146 w^5 + 318 w^6) + S633[a,b,c] (748 + 1474 w - 890 w^2 - 3244 w^3 - 1264 w^4 - 358 w^5 - 138 w^6) + T543[a,b,c] (-564 - 1302 w - 708 w^2 + 1332 w^3 + 2112 w^4 + 780 w^5 + 132 w^6) + U4[a,b,c] (1404 + 3870 w + 3834 w^2 - 3132 w^3 - 6264 w^4 - 2556 w^5 - 396 w^6) h1s2[a_,b_,c_,w_] := S120[a,b,c] (4 - 8 w - 32 w^2) + T111[a,b,c] (-28 + 12 w + 60 w^2 + 64 w^3) + T102[a,b,c] (80 + 6 w - 138 w^2 - 116 w^3 - 48 w^4) + T93[a,b,c] (-108 - 18 w + 126 w^2 + 124 w^3 + 40 w^4 + 16 w^5) + T84[a,b,c] (28 + 12 w + 12 w^2 - 8 w^3 - 48 w^4 + 6 w^5 - 2 w^6) + T75[a,b,c] (136 + 6 w - 186 w^2 - 188 w^3 - 96 w^4 + 6 w^5 - 2 w^6) + S66[a,b,c] (-224 - 20 w + 316 w^2 + 248 w^3 + 80 w^4 + 32 w^5) + S1011[a,b,c] (192 + 228 w - 48 w^2 - 168 w^3 - 96 w^4) + T921[a,b,c] (-472 - 606 w - 114 w^2 + 364 w^3 + 240 w^4 + 48 w^5) + T831[a,b,c] (568 + 1110 w + 450 w^2 - 236 w^3 - 228 w^4 - 36 w^5 - 8 w^6) + T741[a,b,c] (-372 - 1290 w - 534 w^2 + 300 w^3 + 228 w^4 + 66 w^5 - 18 w^6) + T651[a,b,c] (112 + 546 w + 186 w^2 + 124 w^3 + 12 w^4 - 6 w^5 - 2 w^6) + S822[a,b,c] (684 + 1164 w + 84 w^2 - 888 w^3 - 732 w^4 - 300 w^5 - 12 w^6) + T732[a,b,c] (-368 - 1440 w - 252 w^2 + 592 w^3 + 1080 w^4 + 324 w^5 + 64 w^6) + T642[a,b,c] (4 + 702 w + 222 w^2 - 1604 w^3 - 1656 w^4 - 834 w^5 - 74 w^6) + S552[a,b,c] (144 + 348 w - 948 w^2 + 1176 w^3 + 1488 w^4 + 516 w^5 + 192 w^6) + S633[a,b,c] (-32 + 1144 w + 1732 w^2 - 640 w^3 + 56 w^4 - 64 w^5 - 36 w^6) + T543[a,b,c] (-60 - 348 w - 240 w^2 + 1008 w^3 + 744 w^4 + 492 w^5 + 24 w^6) + U4[a,b,c] (576 - 720 w - 864 w^2 - 1728 w^3 - 3996 w^4 - 1044 w^5 - 324 w^6) h1s3[a_,b_,c_,w_] := T111[a,b,c] (-2 + 4 w + 16 w^2) + T102[a,b,c] (12 - 2 w - 14 w^2 - 32 w^3) + T93[a,b,c] (-26 - 9 w + 39 w^2 + 26 w^3 + 24 w^4) + T84[a,b,c] (16 + 2 w - 10 w^2 - 4 w^3 + 4 w^4 - 8 w^5) + T75[a,b,c] (28 + 5 w - 55 w^2 - 26 w^3 + 4 w^4 - 11 w^5 + w^6) + S66[a,b,c] (-56 + 48 w^2 + 72 w^3 + 48 w^4 - 6 w^5 + 2 w^6) + S1011[a,b,c] (4 - 72 w - 84 w^2 - 64 w^3) + T921[a,b,c] (-16 + 37 w + 145 w^2 + 158 w^3 + 72 w^4) + T831[a,b,c] (36 + 11 w - 241 w^2 - 294 w^3 - 182 w^4 - 32 w^5) + T741[a,b,c] (-22 - 27 w + 177 w^2 + 206 w^3 + 90 w^4 + 57 w^5 + 5 w^6) + T651[a,b,c] w (47 - 13 w - 230 w^2 - 122 w^3 - 5 w^4 - w^5) + S822[a,b,c] (144 + 354 w + 66 w^2 - 36 w^3 - 156 w^4 - 48 w^5) + T732[a,b,c] (-292 - 622 w - 478 w^2 + 132 w^3 + 160 w^4 + 118 w^5 + 10 w^6) + T642[a,b,c] (260 + 824 w + 1280 w^2 + 836 w^3 + 184 w^4 - 14 w^5 - 22 w^6) + S552[a,b,c] (-216 - 1182 w - 1326 w^2 - 540 w^3 + 12 w^4 + 12 w^5) + S633[a,b,c] (324 + 702 w + 18 w^2 - 628 w^3 - 456 w^4 - 138 w^5 - 38 w^6) + T543[a,b,c] (-42 - 306 w - 630 w^2 - 540 w^3 - 120 w^4 - 48 w^5 + 12 w^6) + U4[a,b,c] (-312 + 810 w + 2538 w^2 + 2196 w^3 + 972 w^4 + 198 w^5 + 78 w^6) h1[t_,a_,b_,c_,w_] := h1s0[a,b,c,w] + h1s1[a,b,c,w] t + h1s2[a,b,c,w] t^2 + h1s3[a,b,c,w] t^3 h3[t_,a_,b_,c_,w_] := (1+2t)(4-t) h1[(4-t)/(1+2t),a,b,c,w] h2[t_,a_,b_,c_,w_] := t^3 h1[1/t,a,b,c,w] h1f0[a_,b_,c_,w_] := (-16 w + 4) S10[a,b,c] + (24 w^2 + 36 w - 24) T91[a,b,c] + (-12 w^3 - 66 w^2 - 22 w + 52) T82[a,b,c] + (2 w^4 + 10 w^3 + 36 w^2 - 4 w - 32) T73[a,b,c] + (12 w^3 + 66 w^2 + 38 w - 56) T64[a,b,c] + (-4 w^4 - 20 w^3 - 120 w^2 - 64 w + 112) S55[a,b,c] + (-24 w^3 - 108 w^2 - 28 w + 112) S811[a,b,c] + (6 w^4 + 114 w^3 + 216 w^2 - 148 w - 200) T721[a,b,c] + (6 w^4 - 174 w^3 - 180 w^2 + 252 w + 168) T631[a,b,c] + (8 w^4 + 52 w^3 + 48 w^2 - 112 w - 56) T541[a,b,c] + (-96 w^4 - 276 w^3 - 156 w^2 + 476 w + 292) S622[a,b,c] + (104 w^4 + 376 w^3 - 120 w^2 - 508 w - 200) T532[a,b,c] + (-208 w^4 - 236 w^3 + 444 w^2 + 404 w + 112) S442[a,b,c] + (56 w^4 - 200 w^3 - 108 w^2 + 164 w + 64) S433[a,b,c] h1f1[a_,b_,c_,w_] := (-4 w + 28) S10[a,b,c] + (6 w^2 + 8 w - 80) T91[a,b,c] + (24 w^3 - 15 w^2 + 15 w + 66) T82[a,b,c] + (-13 w^4 - 5 w^3 + 18 w^2 - 38 w + 20) T73[a,b,c] + (-10 w^4 - 8 w^3 + 15 w^2 - 11 w - 94) T64[a,b,c] + (6 w^4 + 42 w^3 - 48 w^2 + 60 w + 120) S55[a,b,c] + (-60 w^3 - 186 w^2 + 78 w + 204) S811[a,b,c] + (15 w^4 + 141 w^3 + 300 w^2 - 162 w - 204) T721[a,b,c] + (29 w^4 - 245 w^3 - 330 w^2 + 106 w + 188) T631[a,b,c] + (42 w^4 + 108 w^3 + 114 w^2 - 30 w - 108) T541[a,b,c] + (-138 w^4 - 450 w^3 - 186 w^2 + 342 w + 144) S622[a,b,c] + (138 w^4 + 594 w^3 + 240 w^2 - 342 w - 180) T532[a,b,c] + (-402 w^4 - 798 w^3 - 102 w^2 + 486 w + 348) S442[a,b,c] + (132 w^4 + 48 w^3 - 174 w^2 - 54 w - 60) S433[a,b,c] h1f2[a_,b_,c_,w_] := 2(S2[a,b,c]-S11[a,b,c])( (-8 w + 2) S8[a,b,c] + (12 w^2 + 2 w - 8) T71[a,b,c] + (-6 w^3 - 9 w^2 + w + 8) T62[a,b,c] + (w^4 - 7 w^3 - 12 w^2 - 2 w + 8) T53[a,b,c] + (2 w^4 - 2 w^3 + 18 w^2 + 14 w - 20) S44[a,b,c] + (-12 w^3 - 6 w^2 + 14 w + 46) S611[a,b,c] + (3 w^4 + 21 w^3 + 36 w^2 - 60 w - 36) T521[a,b,c] + (11 w^4 - 5 w^3 + 44 w - 2) T431[a,b,c] + (-36 w^4 - 66 w^3 - 36 w^2 + 24 w + 24) S422[a,b,c] + (4 w^4 + 74 w^3 - 30 w^2 - 14 w + 8) S332[a,b,c]) h1f3[a_,b_,c_,w_] := (8 w - 2) T91[a,b,c] + (-12 w^2 - 2 w + 8) T82[a,b,c] + (6 w^3 + 9 w^2 - w - 8) T73[a,b,c] + (-w^4 + 7 w^3 + 12 w^2 + 2 w - 8) T64[a,b,c] + (-2 w^4 + 2 w^3 - 18 w^2 - 14 w + 20) S55[a,b,c] + (-24 w^2 - 32 w - 4) S811[a,b,c] + (18 w^3 + 57 w^2 + w - 10) T721[a,b,c] + (-4 w^4 - 62 w^3 - 69 w^2 + 7 w + 20) T631[a,b,c] + ( 4 w^4 - 4 w^3 - 6 w^2 + 16 w - 4) T541[a,b,c] + (-6 w^4 - 30 w^3 + 42 w^2 + 106 w + 80) S622[a,b,c] + (22 w^4 + 50 w^3 - 54 w^2 - 170 w - 94) T532[a,b,c] + (-8 w^4 + 80 w^3 + 246 w^2 + 214 w + 32) S442[a,b,c] + (-26 w^4 - 82 w^3 - 120 w^2 + 4 w + 68) S433[a,b,c] h1f[t_,a_,b_,c_,w_] := h1f0[a,b,c,w] + h1f1[a,b,c,w] t + h1f2[a,b,c,w] t^2 + h1f3[a,b,c,w] t^3 h2f[t_,a_,b_,c_,w_] := t^3 h1f[1/t,a,b,c,w] h3f[t_,a_,b_,c_,w_] := (1+2t)(4-t) h1f[(4-t)/(1+2t),a,b,c,w] Factor[h1s0[a,b,c,w] - Delta2[a,b,c,w] h1f0[a,b,c,w]] Factor[h1s1[a,b,c,w] - Delta2[a,b,c,w] h1f1[a,b,c,w]] Factor[h1s2[a,b,c,w] - Delta2[a,b,c,w] h1f2[a,b,c,w]] Factor[h1s3[a,b,c,w] - Delta2[a,b,c,w] h1f3[a,b,c,w]] (*------------------------ Definition of g_i() -----------------------------*) (* p = sigma_1, q = sigma_2, r = sigma_3, pk = p_k^F *) Eliminate[{ss0 == s0[a,b,c], p==a+b+c, q==a b+b c+c a, r==a b c},{a,b,c}] ss0[p_,q_,r_] := (p^6 - 6 p^4 q + 9 p^2 q^2 - 2 q^3 + 6 p^3 r - 12 p q r) ss1[p_,q_,r_] := (p^4 q - 4 p^2 q^2 + 2 q^3 - p^3 r + 7 p q r - 9 r^2) ss2[p_,q_,r_] := (p^2 q^2 - 2 q^3 - 2 p^3 r + 4 p q r - 9 r^2) ss3[p_,q_,r_] := (q^3 - 3 p q r) ss4[p_,q_,r_] := (p^3 r - 3 p q r) ss5[p_,q_,r_] := (p q r - 9 r^2) ss6[p_,q_,r_] := r^2 Factor[ss0[a+b+c, a b+b c+c a, a b c] - s0[a,b,c]] Factor[ss1[a+b+c, a b+b c+c a, a b c] - s1[a,b,c]] Factor[ss2[a+b+c, a b+b c+c a, a b c] - s2[a,b,c]] Factor[ss3[a+b+c, a b+b c+c a, a b c] - s3[a,b,c]] Factor[ss4[a+b+c, a b+b c+c a, a b c] - s4[a,b,c]] Factor[ss5[a+b+c, a b+b c+c a, a b c] - s5[a,b,c]] Factor[ss6[a+b+c, a b+b c+c a, a b c] - s6[a,b,c]] wws[p_,q_,r_] := p0 ss0[p,q,r] + p1 ss1[p,q,r] + p2 ss2[p,q,r] + p3 ss3[p,q,r] + p4 ss4[p,q,r] + p5 ss5[p,q,r] Factor[wws[p,q,r] - (-9(p1+p2+p5)r^2 + ((6p0-p1-2p2+p4)p^3+(-12p0+7p1+4p2-3p3-3p4+p5)p q) r + (p0 p^6+(-6p0+p1)p^4q + (9p0-4p1+p2)p^2q^2 +(-2p0+2p1-2p2+p3)q^3))] (* = 0 *) g0[a_,b_,c_,w_] := -9(p1F[a,b,c,w]+p2F[a,b,c,w]+p5F[a,b,c,w]) g1[Sigma1_,Sigma2_,a_,b_,c_,w_] := (6 p0F[a,b,c,w] - p1F[a,b,c,w] - 2 p2F[a,b,c,w] + p4F[a,b,c,w]) Sigma1^3 + (-12 p0F[a,b,c,w] + 7 p1F[a,b,c,w] + 4 p2F[a,b,c,w] - 3 p3F[a,b,c,w] -3 p4F[a,b,c,w] + p5F[a,b,c,w]) Sigma1 Sigma2 g2[Sigma1_,Sigma2_,a_,b_,c_,w_] := p0F[a,b,c,w] Sigma1^6 + (-6 p0F[a,b,c,w] + p1F[a,b,c,w]) Sigma1^4 Sigma2 + (9 p0F[a,b,c,w] - 4 p1F[a,b,c,w] + p2F[a,b,c,w]) Sigma1^2 Sigma2^2 + (-2 p0F[a,b,c,w] + 2 p1F[a,b,c,w] - 2 p2F[a,b,c,w] + p3F[a,b,c,w]) Sigma2^3 (*------------------------------- g_1(s,t) ---------------------------------*) g1[Sigma1_,Sigma2_,a_,b_,c_,w_] := (6 p0F[a,b,c,w] - p1F[a,b,c,w] - 2 p2F[a,b,c,w] + p4F[a,b,c,w]) Sigma1^3 + (-12 p0F[a,b,c,w] + 7 p1F[a,b,c,w] + 4 p2F[a,b,c,w] - 3 p3F[a,b,c,w] -3 p4F[a,b,c,w] + p5F[a,b,c,w]) Sigma1 Sigma2 g1s0[a_,b_,c_,w_] := (-8w+2) S10[a,b,c] + (12 w^2+18w-12) T91[a,b,c] + (-6w^3-33w^2-11w+26) T82[a,b,c] + (w^4+5w^3+18w^2-2w-16) T73[a,b,c] + (6w^3+33w^2+19w-28) T64[a,b,c] + (-2w^4-10w^3-60w^2-32w+56)S55[a,b,c] + (-12w^3-54w^2-14w+56) S811[a,b,c] + (3w^4+57w^3+108w^2-74w-100) T721[a,b,c] + (3w^4-87w^3-90w^2+126w+84) T631[a,b,c] + (4w^4+26w^3+24w^2-56w-28) T541[a,b,c] + (-48w^4-138w^3-78w^2+238w+146) S622[a,b,c] + (52w^4+188w^3-60w^2-254w-100) T532[a,b,c] + (-104w^4-118w^3+222w^2+202w+56) S442[a,b,c] + (28w^4-100w^3-54w^2+82w+32) S433[a,b,c] g1s1[a_,b_,c_,w_] := (-16w+4) S10[a,b,c] + (24w^2+4w-16) T91[a,b,c] + (-12w^3-18w^2-14w+20) T82[a,b,c] + (2w^4-14w^3) T73[a,b,c] + (4w^4-16w^3+18w^2+30w-24) T64[a,b,c] + (4w^4-28w^3-48w^2-8w+32) S55[a,b,c] + (-24w^3-12w^2+100w+128) S811[a,b,c] + (6w^4+42w^3-12w^2-152w-160) T721[a,b,c] + (22w^4+74w^3+96w^2+224w+88) T631[a,b,c] + (-8w^4+68w^3+72w^2-176w-40) T541[a,b,c] + (-72w^4-156w^3-324w^2+52w-28) S622[a,b,c] + (16w^4+176w^3+96w^2+172w+176) T532[a,b,c] + (-176w^4-556w^3-540w^2-452w-16) S442[a,b,c] + (160w^4+128w^3+372w^2+148w-208) S433[a,b,c] g1s2[a_,b_,c_,w_] := (8w-2) T91[a,b,c] + (-12w^2-2w+8) T82[a,b,c] + (6w^3+9w^2-w-8) T73[a,b,c] + (-w^4+7w^3+12w^2+2w-8) T64[a,b,c] + (-2w^4+2w^3-18w^2-14w+20) S55[a,b,c] + (-24w^2-32w-4) S811[a,b,c] + (18w^3+57w^2+w-10) T721[a,b,c] + (-4w^4-62w^3-69w^2+7w+20) T631[a,b,c] + (4w^4-4w^3-6w^2+16w-4) T541[a,b,c] + (-6w^4-30w^3+42w^2+106w+80) S622[a,b,c] + (22w^4+50w^3-54w^2-170w-94) T532[a,b,c] + (-8w^4+80w^3+246w^2+214w+32) S442[a,b,c] + (-26w^4-82w^3-120w^2+4w+68) S433[a,b,c] Delta6[a_,b_,c_,w_,s_] := g1s2[a,b,c,w] s^2 + g1s1[a,b,c,w] s + g1s0[a,b,c,w] (* Factor[g1[Sigma1,Sigma2,a,b,c,w]] *) Factor[g1[t+2,2t+1,a,b,c,w] - (t+2) Delta2[a,b,c,w] Delta6[a,b,c,w,t]] (* = 0 *) Factor[g1s2[a,b,c,w] Delta6[a,b,c,w,s] - (g1s2[a,b,c,w]s + g1s1[a,b,c,w]/2)^2] Factor[h1[t,a,b,c,w] - (2t g0[a,b,c,w] + g1[t+2,2t+1,a,b,c,w])] (*------------------------------- g_2(s,t) ---------------------------------*) g2[Sigma1_,Sigma2_,a_,b_,c_,w_] := p0F[a,b,c,w] Sigma1^6 + (-6 p0F[a,b,c,w] + p1F[a,b,c,w]) Sigma1^4 Sigma2 + (9 p0F[a,b,c,w] - 4 p1F[a,b,c,w] + p2F[a,b,c,w]) Sigma1^2 Sigma2^2 + (-2 p0F[a,b,c,w] + 2 p1F[a,b,c,w] - 2 p2F[a,b,c,w] + p3F[a,b,c,w]) Sigma2^3 Factor[g2[Sigma1,Sigma2,a,b,c,w]] Delta5[a_,b_,c_,w_] := ((w+1)a-b-c)((w+1)b-c-a)((w+1)c-a-b) (* Virtices of the triangle (1/w:1/w:1/c), (1/a:1/w:1/w), (1/w:1/b:1/w) *) Factor[g2[0,t,a,b,c,w] - t^3(w-1)(a+b+c)^3 Delta5[a,b,c,w] Delta1[a,b,c,w]^2] (* = 0 *) (*-------------------------------- D(p,q) ----------------------------------*) Dpq[Sigma1_,Sigma2_,a_,b_,c_,w_] := g1[Sigma1,Sigma2,a,b,c,w]^2 - 4 g0[a,b,c,w] g2[Sigma1,Sigma2,a,b,c,w] DL0[a_,b_,c_,w_] := (28 w - 4) S10[a,b,c] + (-56 w^2 - 52 w + 24) T91[a,b,c] + (24 w^3 + 128 w^2 + 8 w - 52) T82[a,b,c] + (4 w^4 - 64 w^3 - 32 w^2 + 24 w + 32) T73[a,b,c] + (4 w^5 - 16 w^4 - 24 w^3 - 128 w^2 - 36 w + 56) T64[a,b,c] + (8 w^5 - 40 w^4 + 128 w^3 + 176 w^2 + 56 w - 112) S55[a,b,c] + (120 w^3 + 200 w^2 + 8 w - 112) S811[a,b,c] + (-60 w^4 - 288 w^3 - 256 w^2 + 296 w + 200) T721[a,b,c] + (-20 w^5 + 164 w^4 + 432 w^3 + 80 w^2 - 416 w - 168) T631[a,b,c] + (-16 w^5 + 8 w^4 - 136 w^3 + 32 w^2 + 164 w + 56) T541[a,b,c] + (60 w^5 + 180 w^4 + 288 w^3 - 304 w^2 - 796 w - 292) S622[a,b,c] + (-52 w^5 - 376 w^4 - 232 w^3 + 872 w^2 + 776 w + 200) T532[a,b,c] + (140 w^5 + 284 w^4 - 640 w^3 - 1264 w^2 - 568 w - 112) S442[a,b,c] + (-40 w^5 + 128 w^4 + 680 w^3 - 88 w^2 - 256 w - 64) S433[a,b,c] DL1[a_,b_,c_,w_] := (-8 w - 40) S10[a,b,c] + (16 w^2 + 56 w + 96) T91[a,b,c] + (-48 w^3 - 16 w^2 - 100 w - 52) T82[a,b,c] + (40 w^4 + 8 w^3 + 40 w^2 + 24 w - 40) T73[a,b,c] + (4 w^5 + 20 w^4 + 48 w^3 + 16 w^2 + 108 w + 92) T64[a,b,c] + (8 w^5 - 40 w^4 - 16 w^3 - 112 w^2 - 160 w - 112) S55[a,b,c] + (48 w^3 - 16 w^2 - 280 w - 184) S811[a,b,c] + (-24 w^4 - 72 w^3 - 40 w^2 + 296 w + 56) T721[a,b,c] + (-56 w^5 - 88 w^4 + 72 w^3 + 8 w^2 - 56 w - 24) T631[a,b,c] + (-16 w^5 - 64 w^4 - 208 w^3 + 32 w^2 - 16 w + 56) T541[a,b,c] + (96 w^5 + 288 w^4 + 720 w^3 + 272 w^2 + 176 w + 176) S622[a,b,c] + (-16 w^5 - 376 w^4 - 736 w^3 - 640 w^2 - 520 w - 88) T532[a,b,c] + (248 w^5 + 1112 w^4 + 1520 w^3 + 1328 w^2 + 296 w - 184) S442[a,b,c] + (-184 w^5 - 376 w^4 - 400 w^3 - 304 w^2 + 392 w + 152) S433[a,b,c] DL2[a_,b_,c_,w_] := (16 w + 8) S10[a,b,c] + (-32 w^2 - 40 w - 12) T91[a,b,c] + (24 w^3 + 68 w^2 + 29 w - 13) T82[a,b,c] + (-8 w^4 - 34 w^3 - 26 w^2 + 6 w + 26) T73[a,b,c] + (w^5 - 13 w^4 - 24 w^3 - 68 w^2 - 45 w + 5) T64[a,b,c] + (2 w^5 - 10 w^4 + 68 w^3 + 116 w^2 + 68 w - 28) S55[a,b,c] + (48 w^3 + 104 w^2 + 74 w - 10) S811[a,b,c] + (-24 w^4 - 126 w^3 - 118 w^2 + 74 w + 86) T721[a,b,c] + (4 w^5 + 104 w^4 + 198 w^3 + 38 w^2 - 194 w - 78) T631[a,b,c] + (-4 w^5 + 20 w^4 - 16 w^3 + 8 w^2 + 86 w + 14) T541[a,b,c] + (6 w^5 + 18 w^4 - 36 w^3 - 220 w^2 - 442 w - 190) S622[a,b,c] + (-22 w^5 - 94 w^4 + 68 w^3 + 596 w^2 + 518 w + 122) T532[a,b,c] + (8 w^5 - 136 w^4 - 700 w^3 - 964 w^2 - 358 w - 10) S442[a,b,c] + (26 w^5 + 158 w^4 + 440 w^3 + 32 w^2 - 226 w - 70) S433[a,b,c] DL[t_,a_,b_,c_,w_] := DL2[a,b,c,w] t^2 + DL1[a,b,c,w] t + DL0[a,b,c,w] Df[t_,a_,b_,c_,w_] := Dpq[t+2,2t+1,a,b,c,w] (*---------------------------- Proposition 3.8 -----------------------------*) (*--------------------------- Proposition 3.8(1) ---------------------------*) Factor[Delta3[x,1,1,w]] (* = (x-1)^2(x-w)^2 >= 0 *) Factor[Delta3[1,0,0,w]] (* = 1 *) (*--------------------------- Proposition 3.8(2) ---------------------------*) g0[a_,b_,c_,w_] := -9(p1F[a,b,c,w]+p2F[a,b,c,w]+p5F[a,b,c,w]) Factor[g0[a,b,c,w] - 9(a+b+c)^2 (S2[a,b,c]-S11[a,b,c]) Delta2[a,b,c,w]^2 Delta3[a,b,c,w]] (* = 0 *) (* Delta3[a,b,c,w] is PSD. Thus $g0[a,b,c,w] is PSD +) (*--------------------------- Proposition 3.8(3) ---------------------------*) Factor[Df[t,a,b,c,w] - (w-1) Delta1[a,b,c,w]^2 Delta2[a,b,c,w]^2 ((2t+1)S2[a,b,c]-(t^2+2) S11[a,b,c])^2 DL[t,a,b,c,w]] (* = 0 *) (*--------------------------- Proposition 3.8(4) ---------------------------*) Factor[Df[1,a,b,c,w] - 81(w-1)^2(S2[a,b,c]-S11[a,b,c])^2 Delta1[a,b,c,w]^2 Delta2[a,b,c,w]^2 Delta4[a,b,c,w]^2] (* Df[1,a,b,c,w] \geq 0 *) (*--------------------------- Proposition 3.8(5) ---------------------------*) Factor[Df[-2,a,b,c,w] - 972(w-1)(a+b+c)^5 (S2[a,b,c]-S11[a,b,c]) Delta1[a,b,c,w]^2 Delta2[a,b,c,w]^2 Delta3[a,b,c,w] Delta5[a,b,c,w]] (* = 0 *) (*--------------------------- Proposition 3.8(6) ---------------------------*) Factor[h1[-2,a,b,c,w] + 4 g0[a,b,c,w]] (* = 0 *) Factor[h1[-2,a,b,c,w] + 36 9(a+b+c)^2 (S2[a,b,c]-S11[a,b,c]) Delta2[a,b,c,w]^2 Delta3[a,b,c,w]] (* = 0 *) (* Thus h1[-2,a,b,c,w] <= 0 *) (*--------------------------- Proposition 3.8(7) ---------------------------*) Factor[h1[1,a,b,c,w] - 9(1-w)(S2[a,b,c]-S11[a,b,c]) Delta1[a,b,c,w] Delta2[a,b,c,w] Delta4[a,b,c,w]] (*--------------------------- Proposition 3.8(8) ---------------------------*) Factor[FrakF[0,-1,1,u,v,1,w] - (1-w) (u+v+1)^3 Delta1[u,v,1,w]^2 Delta5[u,v,1,w]] (* = 0 *) (* Cf. *) Factor[p0F[0,-1,1,w]] (* = 0 *) Factor[p1F[0,-1,1,w]] (* = 0 *) Factor[p2F[0,-1,1,w]] (* = 0 *) Factor[p3F[0,-1,1,w]] (* = 0 *) Factor[p4F[0,-1,1,w]] (* = 0 *) Factor[p5F[0,-1,1,w]] (* = 0 *) Factor[Delta1[0,-1,1,w]] (* = 0 *) Factor[Delta2[0,-1,1,w]] (* = (w+2)^0 *) Factor[Delta3[0,-1,1,w]] (* = (w+2)^0 *) Factor[Delta4[0,-1,1,w]] (* = 0 *) Factor[Delta5[0,-1,1,w]] (* = 0 *) (* p0F[x,y-1,1,w] = (w+2)^2 (4(w+1)(11w+13)x^2 + 8(w+1)(2w-5)x y - (w^2+12w-16)y^2) + (higher terms w.r.t (x,y)) *) (*-------------------------- Proposition 3.8(Misc) -------------------------*) Delta4[a_,b_,c_,w_] := 2 S5[a,b,c] - (2w+3) T41[a,b,c] + (-w^2+2w+1) T32[a,b,c] + 4(w+1)^2 US2[a,b,c] - (2w^2+8w+2) US11[a,b,c] Factor[DL2[a,b,c,w] DL[t,a,b,c,w] - ((DL2[a,b,c,w] t + DL1[a,b,c,w]/2)^2 + 108(w-1) (a+b+c)(S2[a,b,c]-S11[a,b,c]) Delta3[a,b,c,w] Delta4[a,b,c,w]^2 Delta5[a,b,c,w])] (* = 0 *) DLmin[t_,a_,b_,c_,w_] := 2t DL2[a,b,c,w] + DL1[a,b,c,w] Factor[DLmin[-2,a,b,c,w] - (-72)(a+b+c)(S2[a,b,c]-S11[a,b,c])Delta3[a,b,c,w] Delta5[a,b,c,w]] (* = 0 *) Factor[DLmin[1,a,b,c,w] - 6(w-1) Delta4[a,b,c,w]^2] (* = 0 *) Factor[h3[-2,a,b,c,w] - 648(a+b+c)^2 (S2[a,b,c]-S11[a,b,c]) Delta2[a,b,c,w]^2 Delta3[a,b,c,w]] (* Factor[h3[-2,a,b,c,w] >= 0 *) ContourPlot[{p0F[x,y,1,2]==0, Delta1[x,y,1,2]==0, Delta2[x,y,1,2]==0, Delta5[x,y,1,2]==0}, {x,-2,3}, {y,-2,3}] ContourPlot[{p0F[x,y,1,3]==0, Delta1[x,y,1,3]==0, Delta2[x,y,1,3]==0, Delta5[x,y,1,3]==0}, {x,-2,3}, {y,-2,3}] ContourPlot[{p0F[x,y,1,-1]==0, Delta1[x,y,1,-1]==0, Delta2[x,y,1,-1]==0, Delta5[x,y,1,-1]==0}, {x,-2,3}, {y,-2,3}] ContourPlot[{p0F[x,y,1,-3]==0, Delta1[x,y,1,-3]==0, Delta2[x,y,1,-3]==0, Delta5[x,y,1,-3]==0}, {x,-2,3}, {y,-2,3}] (*---------------------------- Proposition 3.9 ------------------------------*) Delta1[a_,b_,c_,w_] := (2 S3[a,b,c] - (w+2) T21[a,b,c] + (6w+6) U[a,b,c]) Delta2[a_,b_,c_,w_] := ((2w+1)S2[a,b,c] - (w^2+2)S11[a,b,c]) Xi[a_,b_,c_,w_] := (a+b+c) (1-w) Delta1[a,b,c,w] Delta2[a,b,c,w] fw0[a_,b_,c_,w_] := (4 S120[a,b,c] + (-8 w - 12) T111[a,b,c] + (5 w^2 + 28 w + 11) T102[a,b,c] + (-w^3 - 15 w^2 - 24 w - 8) T93[a,b,c] + (2 w^3 + 10 w^2 - 16 w + 8) T84[a,b,c] + (w^3 + 15 w^2 + 32 w + 20) T75[a,b,c] + (-4 w^3 - 30 w^2 - 24 w - 46) S66[a,b,c] + (14 w^2 + 40 w + 38) S1011[a,b,c] + (-7 w^3 - 57 w^2 - 96 w - 32) T921[a,b,c] + (w^4 + 16 w^3 + 101 w^2 + 120 w + 10) T831[a,b,c] + (w^4 - 34 w^3 - 69 w^2 - 64 w - 38) T741[a,b,c] + (-2 w^4 + 25 w^3 + 11 w^2 + 8 w + 34) T651[a,b,c] + (2 w^4 + 60 w^3 + 138 w^2 + 112 w + 24) S822[a,b,c] + (-13 w^4 - 87 w^3 - 186 w^2 - 88 w + 6) T732[a,b,c] + (26 w^4 + 88 w^3 + 133 w^2 + 52 w - 3) T642[a,b,c] + (-26 w^4 - 108 w^3 - 66 w^2 - 16 w - 12) S552[a,b,c] + (12 w^4 + 102 w^3 + 132 w^2 + 24 w + 34) S633[a,b,c] + (-12 w^4 - 38 w^3 - 32 w^2 - 32 w - 42) T543[a,b,c] + (30 w^4 + 60 w^3 - 60 w^2 + 120 w + 150) U4[a,b,c]) fw1[a_,b_,c_,w_] := (-4 T111[a,b,c] + (-8 w^2 + 6 w + 14) T102[a,b,c] + (8 w^3 + 2 w^2 - 14 w - 12) T93[a,b,c] + (-2 w^4 - 2 w^3 + 12 w^2 + 4 w - 8) T84[a,b,c] + (-8 w^3 - 2 w^2 + 14 w + 16) T75[a,b,c] + (4 w^4 + 4 w^3 - 8 w^2 - 20 w - 12) S66[a,b,c] + (16 w^2 + 20 w + 20) S1011[a,b,c] + (-8 w^3 - 26 w^2 - 82 w - 48) T921[a,b,c] + (-2 w^3 + 60 w^2 + 154 w + 60) T831[a,b,c] + (4 w^4 - 22 w^3 - 120 w^2 - 106 w - 32) T741[a,b,c] + (-8 w^4 + 32 w^3 + 70 w^2 + 14 w + 4) T651[a,b,c] + (4 w^4 + 40 w^3 + 144 w^2 + 164 w + 56) S822[a,b,c] + (-8 w^4 - 66 w^3 - 238 w^2 - 228 w - 44) T732[a,b,c] + (42 w^4 + 166 w^3 + 272 w^2 + 174 w + 26) T642[a,b,c] + (-28 w^4 - 232 w^3 - 288 w^2 - 68 w - 8) S552[a,b,c] + (-28 w^4 + 76 w^3 + 292 w^2 + 144 w + 12) S633[a,b,c] + (-24 w^4 - 112 w^3 - 140 w^2 - 56 w - 16) T543[a,b,c] + (120 w^4 + 420 w^3 + 240 w^2 + 60) U4[a,b,c]) fw[x_,a_,b_,c_,w_] := p0F[a,b,c,w] x^2 + fw1[a,b,c,w] x + fw0[a,b,c,w] Factor[FrakF[x,1,1,u,v,1,w] - (x-1)^2(x-w)^2 fw[x,u,v,1,w] ] (* = 0 *) Factor[p0F[u,v,1,w] FrakF[x,1,1,u,v,1,w] - (x-1)^2(x-w)^2 ((p0F[u,v,1,w] x + fw1[u,v,1,w]/2)^2 + (u-v)^4(v-1)^4(1-u)^4(u+v+1)^3 (1-w) Delta1[u,v,1,w] Delta2[u,v,1,w]^3)] (* = 0 *) (*----------------------------- Remark 3.11 ---------------------------------*) Factor[FrakF[a,b,c,u,v,1,1] - (S2[u,v,1]-S11[u,v,1])^3 ((T21[u,v,1]-6U[u,v,1])S3[a,b,c] - (S3[u,v,1]-3U[u,v,1])T21[a,b,c] + (6 S3[u,v,1]-3 T21[u,v,1]) U[a,b,c])^2] (*-------------------------- Proof of Theorem 1.5 ---------------------------*) p0F[-1/2,-1/3,1,9/10] (* = 2838188587/147622500 *) Xi[-1/2,-1/3,1,9/10] (* = 461719/911250 *) Delta1[-1/2,-1/3,1,9/10] (* = 722/135 *) Delta2[-1/2,-1/3,1,9/10] (* = 1279/225 *) Delta4[-1/2,-1/3,1,9/10] (* = 255823/24300 *) Factor[h1[t,-1/2,-1/3,1,9/10]] (* = (1279/132860250000)(-4763259726 t^3 + 10555137817 t^2 + 53854835215 t + 1028618365). Roots: -2.419724773331076`, -0.019172500257541247`, 4.654845683746759` *) Factor[DL[t,-1/2,-1/3,1,9/10]] (* = (1/5904900000)(398926806344 t^2 - 1190526056662 t + 202590584357). Roots: (595263028331 - 767469 Sqrt[464372213673])/398926806344 = 0.1811669227511762 *) Plot[h1[t,-1/2,-1/3,1,9/10], {t,-2,1}] Plot[Df[t,-1/2,-1/3,1,9/10], {t,-2,1}] Plot[DL[t,-1/2,-1/3,1,9/10], {t,-2,1}] Plot3D[FrakF[x,y,1,-1/2,-1/3,1,9/10], {x,-1,1}, {y,-1,1}] ContourPlot[FrakF[x,y,1,-1/2,-1/3,1,9/10]== 0, {x,-2,2}, {y,-2,2}] (*===========================================================================*) (*============================== Section 4 ==================================*) (*===========================================================================*) S1[a_, b_, c_] := a + b + c S2[a_, b_, c_] := a^2 + b^2 + c^2 S11[a_, b_, c_] := a b + b c + c a S3[a_, b_, c_] := a^3 + b^3 + c^3 S21[a_, b_, c_] := a^2b + b^2c + c^2a S12[a_, b_, c_] := a b^2 + b c^2 + c a^2 T21[a_, b_, c_] := S21[a,b,c] + S12[a,b,c] U[a_, b_, c_] := a b c S4[a_,b_,c_] := (a^4+b^4+c^4) S31[a_,b_,c_] := (a^3b + b^3c + c^3a) S13[a_,b_,c_] := (a b^3 + b c^3 + c a^3) S22[a_,b_,c_] := (a^2b^2 + b^2c^2 + c^2a^2) US1[a_,b_,c_] := a b c(a + b + c) T31[a_,b_,c_] := S31[a,b,c] + S13[a,b,c] S5[a_, b_, c_] := a^5 + b^5 + c^5 S41[a_, b_, c_] := a^4 b + b^4 c + c^4 a S14[a_, b_, c_] := a b^4 + b c^4 + c a^4 S32[a_, b_, c_] := a^3 b^2 + b^3 c^2 + c^3 a^2 S23[a_, b_, c_] := a^2 b^3 + b^2 c^3 + c^2 a^3 T41[a_, b_, c_] := S41[a,b,c] + S14[a,b,c] T32[a_, b_, c_] := S32[a,b,c] + S23[a,b,c] US2[a_, b_, c_] := a b c(a^2+b^2+c^2) US11[a_, b_, c_] := a b c(a b+b c+c a) s0[a_,b_,c_] := S5[a,b,c]-US11[a,b,c] s1[a_,b_,c_] := T41[a,b,c]-2US11[a,b,c] s2[a_,b_,c_] := T32[a,b,c]-2US11[a,b,c] s3[a_,b_,c_] := US2[a,b,c]-US11[a,b,c] s4[a_,b_,c_] := US11[a,b,c] s[a_,b_,c_] := {s0[a,b,c], s1[a,b,c], s2[a,b,c], s3[a,b,c], s4[a,b,c]} p1A[t_,u_] := (u^2 - (t+2)(5t^2+t+9) u + 9(t-1)^2(t+2)^2)/((5t+1)(t+2)u) p2A[t_,u_] := (-t^2 u^3 + (t-1)(7t^3-t^2+11t+1)u^2 + (t+2)(17t^5-25t^4+199t^3-59t^2+76t+8)u + 9(t-1)^4(t+2)^2(t^2-12t-1))/((5t+1)^3(t+2)u) p3A[t_,u_]:=((2t^3+4t^2+5t+1)u^3 - 2(t+2)(7t^4+42t^3+37t^2+48t+10)u^2 + (t+2)^2(91t^5+125t^4+682t^3+182t^2+523t+125)u - 18(t-1)^2(t+2)^3(t^4+36t^3+34t^2+60t+13))/((t+2)^2(5t+1)^3u) p4A[t_,u_]:=((t-1)^3(6t^2+6t-12 + u)^3)/((t+2)^2(5t+1)^3u) FrakEA[a_,b_,c_,t_,u_] := s0[a,b,c] + p1A[t,u] s1[a,b,c] + p2A[t,u] s2[a,b,c] + p3A[t,u] s3[a,b,c] + p4A[t,u] s4[a,b,c] (* $0 \leq t \leq 7$, $t \ne 1$, $0 \leq u \leq \mu_A(t)$, *) muH[t_] := (t+2)(7-t) muL[t_] := 9 (t-1)^2 muA[t_] := If[t<5/2, muL[t], muH[t]] muZ[t_,u_] := ((t+2)(7-t) - u)/((t+2)(5t+1)) p1B[t_,w_] := -2w-3 p2B[t_,w_] := w^2 + 2w + 2 p3B[t_,w_] := -((2t^3+4t^2+5t+1)/(t^2(t+2))) w^2 + (2(4t^2+5t+3)/(t+2)) w - ((3t^3-7t^2-12t-8)/(t+2)) p4B[t_,w_] := (t-1)^3(-w^2 - 2t^2w +t^2(t-2))/(t^2(t+2)) FrakEB[a_,b_,c_,t_,u_] := s0[a,b,c] + p1B[t,u+1/u-2] s1[a,b,c] + p2B[t,u+1/u-2] s2[a,b,c] + p3B[t,u+1/u-2] s3[a,b,c] + p4B[t,u+1/u-2] s4[a,b,c] (* $t \geq 2$, $s_m(t) \leq u < 1$ *) muR[t_] := 2-t^2 + t Sqrt[(t-1)(t+2)] muB[t_] := (muR[t] - Sqrt[muR[t]^2 - 4])/2 FrakEC[a_,b_,c_,t_] := s0[a,b,c] - (t+1) s1[a,b,c] + t s2[a,b,c] + (t+1)^2 s3[a,b,c] (* $0 \leqq t \leqq 2$ *) FrakED[a_,b_,c_,t_] := s1[a,b,c] + (t^2-1) s2[a,b,c] - 2(t+1)^2 s3[a,b,c] (* $t geqq 0$ *) p3E[t_] := -(4t^2+5t+3)/(t+2) p4E[t_] := (t-1)^3/(t+2) FrakEE[a_,b_,c_,t_] := s1[a,b,c]-s2[a,b,c] + p3E[t] s3[a,b,c] + p4E[t] s4[a,b,c] sa0[a_,b_,c_] := 5 a^4 - 2 a b^2 c - 2 a b c^2 - b^2 c^2 sa1[a_,b_,c_] := (b + c) (4 a^3 + b^3 - 4 a b c - b^2 c - b c^2 + c^3) sa2[a_,b_,c_] := 3a^2b^2+2a b^3-4a b^2c+3a^2c^2-4a b c^2-2b^2c^2+2a c^3 sa3[a_,b_,c_] := b c (3 a^2 - 2 a b + b^2 - 2 a c - b c + c^2) sa4[a_,b_,c_] := b c (2 a b + 2 a c + b c) sa[a_,b_,c_] := {sa0[a,b,c], sa1[a,b,c], sa2[a,b,c], sa3[a,b,c], sa4[a,b,c]} sb0[a_,b_,c_] := 5 b^4 - 2 a^2 b c - a^2 c^2 - 2 a b c^2 sb1[a_,b_,c_] := (a + c) (a^3 + 4 b^3 - a^2 c - 4 a b c - a c^2 + c^3) sb2[a_,b_,c_] := 2a^3b+3a^2b^2-4a^2b c-2a^2c^2-4a b c^2+3b^2c^2+2b c^3 sb3[a_,b_,c_] := a c (a^2 - 2 a b + 3 b^2 - a c - 2 b c + c^2) sb4[a_,b_,c_] := a c (2 a b + a c + 2 b c) sb[a_,b_,c_] := {sb0[a,b,c], sb1[a,b,c], sb2[a,b,c], sb3[a,b,c], sb4[a,b,c]} sc0[a_,b_,c_] := -a^2 b^2 - 2 a^2 b c - 2 a b^2 c + 5 c^4 sc1[a_,b_,c_] := (a + b) (a^3 - a^2 b - a b^2 + b^3 - 4 a b c + 4 c^3) sc2[a_,b_,c_] := -2a^2b^2+2a^3c-4a^2b c-4a b^2c+2b^3c+3a^2c^2+3b^2c^2 sc3[a_,b_,c_] := a b (a^2 - a b + b^2 - 2 a c - 2 b c + 3 c^2) sc4[a_,b_,c_] := a b (a b + 2 a c + 2 b c) sc[a_,b_,c_] := {sc0[a,b,c], sc1[a,b,c], sc2[a,b,c], sc3[a,b,c], sc4[a,b,c]} saa0[a_,b_,c_] := 2 (10 a^3 - b^2 c - b c^2) saa1[a_,b_,c_] := 4 (b + c) (3 a^2 - b c) saa2[a_,b_,c_] := 2 (3 a b^2 + b^3 - 2 b^2 c + 3 a c^2 - 2 b c^2 + c^3) saa3[a_,b_,c_] := 2 b c (3 a - b - c) saa4[a_,b_,c_] := 2 b c (b + c) saa[a_,b_,c_] := {saa0[a,b,c], saa1[a,b,c], saa2[a,b,c], saa3[a,b,c], saa4[a,b,c]} sbb[a_,b_,c_] := saa[b,a,c] scc[a_,b_,c_] := saa[c,a,b] sab0[a_,b_,c_] := -2 c (2 a b + a c + b c) sab1[a_,b_,c_] := 4 (a^3 + b^3 - 2 a b c - a c^2 - b c^2) sab2[a_,b_,c_] := 2 (3 a^2 b + 3 a b^2 - 4 a b c - 2 a c^2 - 2 b c^2) sab3[a_,b_,c_] := c (3 a^2 - 4 a b + 3 b^2 - 2 a c - 2 b c + c^2) sab4[a_,b_,c_] := 2 c (2 a b + a c + b c) sab[a_,b_,c_] := {sab0[a,b,c], sab1[a,b,c], sab2[a,b,c], sab3[a,b,c], sab4[a,b,c]} saaa0[a_,b_,c_] := 60a^2 saaa1[a_,b_,c_] := 24 a (b + c) saaa2[a_,b_,c_] := 6 (b^2 + c^2) saaa3[a_,b_,c_] := 6 b c saaa4[a_,b_,c_] := 0 saaa[a_,b_,c_] := {saaa0[a,b,c], saaa1[a,b,c], saaa2[a,b,c], saaa3[a,b,c], saaa4[a,b,c]} saaaa0[a_,b_,c_] := 120a saaaa1[a_,b_,c_] := 24(b + c) saaaa2[a_,b_,c_] := 0 saaaa3[a_,b_,c_] := 0 saaaa4[a_,b_,c_] := 0 saaaa[a_,b_,c_] := {saaaa0[a,b,c], saaaa1[a,b,c], saaaa2[a,b,c], saaaa3[a,b,c], saaaa4[a,b,c]} (*============================= Section 4.1 =================================*) (============================= Section 4.1.2 ================================*) (*----------------------------- Theorem 4.3 ---------------------------------*) (*---------------------------- Theorem 4.3(1) -------------------------------*) (* ${\frac e}_t^C$ *) FrakEC[a_,b_,c_,t_] := s0[a,b,c] - (t+1) s1[a,b,c] + t s2[a,b,c] + (t+1)^2 s3[a,b,c] Factor[FrakEC[x,1,1,t] - x(x-1)^2(x-t)^2] (* = 0 *) Factor[FrakEC[0,x,1,t] - (x-1)^2(x+1)((x-1)^2+(2-t)x)] (* = 0 *) (*---------------------------- Theorem 4.3(2) -------------------------------*) (* (2) Case $0 muL[t] \leq muH[t], 1 < (muH[t]-muL[t])/((t+2)(5t+1)) \leq muH[t]/((t+2)(5t+1)). muH[5/2]=muL[5/2]= = 20. 5/2 < t \leq 7$ ==> muL[t] \geq muH[t], 0 \leq muZ[t,u] \leq muH[t]/((t+2)(5t+1)) < 1 *) (* muZ[t,muL[t]] = (muH[t]-muL[t])/((t+2)(5t+1)) = (5-2t)/(t+2) *) Solve[muZ[t,u] == v, u] (* u = -(t+2)(5t v+t+v-7) *) Factor[FrakEA[x,1,1,t,u]] Factor[FrakEA[x,1,1,t,-(t+2)(5t x+t+x-7)]] (* = 0 *) (*---------------------------- Theorem 4.7(3) -------------------------------*) (* Case $t=1$, $u=0$, muZ[1,0]=1. *) Factor[FrakEA0[a,b,c,1] - 36 (s1[a,b,c] - 8 s3[a,b,c])] (* = 0 *) Factor[FrakEA0[a,b,c,1]] (* = 36(S2[a,b,c]-S11[a,b,c])(T21[a,b,c]-6U[a,b,c]) *) AA1 := {s[1,1,1], saa[1,1,1], saaa[1,1,1], s[0,0,1]} AA1 Factor[NullSpace[AA1]] (*---------------------------- Theorem 4.7(4) -------------------------------*) (* $\fre_{7,0}^A = 1296 \fre_{0,0}^A$. *) Factor[FrakEA0[a,b,c,7] - 1296FrakEA0[a,b,c,0]] Factor[FrakEA0[a,b,c,7] - 1296 FrakEE[a,b,c,7]] (* = 0 *) pA1n[t_,u_] := (u^2 - (t+2)(5t^2+t+9) u + 9(t-1)^2(t+2)^2)/((5t+1)(t+2)) pA2n[t_,u_] := (-t^2 u^3 + (t-1)(7t^3-t^2+11t+1)u^2 + (t+2)(17t^5-25t^4+199t^3-59t^2+76t+8)u + 9(t-1)^4(t+2)^2(t^2-12t-1))/((5t+1)^3(t+2)) pA3n[t_,u_]:=((2t^3+4t^2+5t+1)u^3 - 2(t+2)(7t^4+42t^3+37t^2+48t+10)u^2 + (t+2)^2(91t^5+125t^4+682t^3+182t^2+523t+125)u - 18(t-1)^2(t+2)^3(t^4+36t^3+34t^2+60t+13))/((t+2)^2(5t+1)^3) pA4n[t_,u_]:=((t-1)^3(6t^2+6t-12 + u)^3)/((t+2)^2(5t+1)^3) FrakEAn[a_,b_,c_,t_,u_] := u s0[a,b,c]+pA1n[t,u] s1[a,b,c]+pA2n[t,u] s2[a,b,c]+ pA3n[t,u] s3[a,b,c]+pA4n[t,u] s4[a,b,c] Factor[FrakEAn[a,b,c,7,0] - (9/2)FrakEAn[a,b,c,0,0]] (*------------------------------ Remark 4.8 ---------------------------------*) (*----------------------------- Remark 4.8(1) -------------------------------*) Factor[FrakEA[a,b,c,t,3(t-1)^2(t+2)/(2t+1)] - (S2[a,b,c] - (S2[t,1,1]/S11[t,1,1])S11[a,b,c])^2] (*============================ Section 4.1.6 ================================*) (*----------------------------- Theorem 4.10 --------------------------------*) (* $\fre_{t,u}^B$, $t \geq 2$, $\mu_B(t) \leq u \leq 1$ *) muR[t_] := 2-t^2 + t Sqrt[(t-1)(t+2)] muB[t_] := (muR[t] - Sqrt[muR[t]^2 - 4])/2 p1B[t_,w_] := -2w-3 p2B[t_,w_] := w^2 + 2w + 2 p3B[t_,w_] := -((2t^3+4t^2+5t+1)/(t^2(t+2))) w^2 + (2(4t^2+5t+3)/(t+2)) w - ((3t^3-7t^2-12t-8)/(t+2)) p4B[t_,w_] := (t-1)^3(-w^2 - 2t^2w +t^2(t-2))/(t^2(t+2)) FrakEB[a_,b_,c_,t_,u_] := s0[a,b,c] + p1B[t,u+1/u-2] s1[a,b,c] + p2B[t,u+1/u-2] s2[a,b,c] + p3B[t,u+1/u-2] s3[a,b,c] + p4B[t,u+1/u-2] s4[a,b,c] (*-------------------------- Theorem 4.10(0-i) ------------------------------*) Factor[FrakEB[0,x,1,t,u] - (x+1)(x-u)^2(x-1/u)^2] (* = 0 *) ftwB[a_,b_,c_,t_,w_] := s0[a,b,c] + pB1[t,w] s1[a,b,c] + pB2[t,w] s2[a,b,c] + pB3[t,w] s3[a,b,c] + pB4[t,w] s4[a,b,c] c0B[t_,w_] := t^2(t+2) c1B[t_,w_] := 2t^2(t+2)(-2w+t-3) (* Zm[t]>(t-3)/2. c1<0 if (t-3)/2 OK *) muR[t_] := 2-t^2 + t Sqrt[(t-1)(t+2)] muB[t_] := (muR[t] - Sqrt[muR[t]^2 - 4])/2 b1[t_,z_] := (2t+1)z - t(t-4) (* p = Sqrt[t], q = Sqrt[s] *) ABf2[p_,q_]:={f[p,1,1], f[-p,1,1], f[p,-1,1], f[p,1,-1], f[1,p,1], f[-1,p,1], f[1,-p,1], f[1,p,-1], f[1,1,p], f[-1,1,p], f[1,-1,p], f[1,1,-p], f[0,q,1], f[0,1,q], f[q,0,1], f[q,1,0], f[1,0,q], f[1,q,0], f[0,-q,1], f[0,1,-q], f[-q,0,1], f[-q,1,0], f[1,0,-q], f[1,-q,0]} WBf2[p_,q_]:={f[p,1,1], f[-p,1,1], f[p,-1,1], f[p,1,-1], f[1,p,1], f[-1,p,1], f[1,-p,1], f[1,p,-1], f[1,1,p], f[-1,1,p], f[1,-1,p], f[q,1,0], f[q,0,1], f[-q,1,0], f[-q,0,1], f[1,q,0], f[1,0,q], f[1,0,-q], f[0,-q,1], f[0,1,q], f[0,1,-q]} Factor[Det[WBf2[p,q]] + 16384 p^6(p^2-1)^7(p^2+2) q^8(q-1)^5(q+1)^4(q^2+1)^4 ((q+1)^2+p^2q)(q^2(p^2+1)-1)(p^4q^2-2p^2q^4-2p^2-(q^2-1)^2)^3] (* q^2(p^2+1)-1 == 0 ==> s == 1/(t+1) < muB[t] *) rp[p_,q_] := p^4q^2-2p^2q^4-2p^2-(q^2-1)^2 rs[s_,t_] := s t^2-2s^2t-2t-s^2+2s-1 (* z = s+1/2-2, rs[s,t]/s == t^2-4t - z(2t+1) == -b1[t,z] *) (*---------------------------- Theorem 4.28 ---------------------------------*) f1[x_,y_,z_]:=x^10 f2[x_,y_,z_]:=x^9y f3[x_,y_,z_]:=x^9z f4[x_,y_,z_]:=x^8y^2 f5[x_,y_,z_]:=x^8y z f6[x_,y_,z_]:=x^8z^2 f7[x_,y_,z_]:=x^7y^3 f8[x_,y_,z_]:=x^7y^2z f9[x_,y_,z_]:=x^7y z^2 f10[x_,y_,z_]:=x^7z^3 f11[x_,y_,z_]:=x^6y^4 f12[x_,y_,z_]:=x^6y^3z f13[x_,y_,z_]:=x^6y^2z^2 f14[x_,y_,z_]:=x^6y z^3 f15[x_,y_,z_]:=x^6z^4 f16[x_,y_,z_]:=x^5y^5 f17[x_,y_,z_]:=x^5y^4z f18[x_,y_,z_]:=x^5y^3z^2 f19[x_,y_,z_]:=x^5y^2z^3 f20[x_,y_,z_]:=x^5y z^4 f21[x_,y_,z_]:=x^5z^5 f22[x_,y_,z_]:=x^4y^6 f23[x_,y_,z_]:=x^4y^5z f24[x_,y_,z_]:=x^4y^4z^2 f25[x_,y_,z_]:=x^4y^3z^3 f26[x_,y_,z_]:=x^4y^2z^4 f27[x_,y_,z_]:=x^4y z^5 f28[x_,y_,z_]:=x^4z^6 f29[x_,y_,z_]:=x^3y^7 f30[x_,y_,z_]:=x^3y^6z f31[x_,y_,z_]:=x^3y^5z^2 f32[x_,y_,z_]:=x^3y^4z^3 f33[x_,y_,z_]:=x^3y^3z^4 f34[x_,y_,z_]:=x^3y^2z^5 f35[x_,y_,z_]:=x^3y z^6 f36[x_,y_,z_]:=x^3z^7 f37[x_,y_,z_]:=x^2y^8 f38[x_,y_,z_]:=x^2y^7z f39[x_,y_,z_]:=x^2y^6z^2 f40[x_,y_,z_]:=x^2y^5z^3 f41[x_,y_,z_]:=x^2y^4z^4 f42[x_,y_,z_]:=x^2y^3z^5 f43[x_,y_,z_]:=x^2y^2z^6 f44[x_,y_,z_]:=x^2y z^7 f45[x_,y_,z_]:=x^2z^8 f46[x_,y_,z_]:=x y^9 f47[x_,y_,z_]:=x y^8z f48[x_,y_,z_]:=x y^7z^2 f49[x_,y_,z_]:=x y^6z^3 f50[x_,y_,z_]:=x y^5z^4 f51[x_,y_,z_]:=x y^4z^5 f52[x_,y_,z_]:=x y^3z^6 f53[x_,y_,z_]:=x y^2z^7 f54[x_,y_,z_]:=x y z^8 f55[x_,y_,z_]:=x z^9 f56[x_,y_,z_]:=y^10 f57[x_,y_,z_]:=y^9z f58[x_,y_,z_]:=y^8z^2 f59[x_,y_,z_]:=y^7z^3 f60[x_,y_,z_]:=y^6z^4 f61[x_,y_,z_]:=y^5z^5 f62[x_,y_,z_]:=y^4z^6 f63[x_,y_,z_]:=y^3z^7 f64[x_,y_,z_]:=y^2z^8 f65[x_,y_,z_]:=y z^9 f66[x_,y_,z_]:=z^10 f[x_,y_,z_]:={f1[x,y,z],f2[x,y,z],f3[x,y,z],f4[x,y,z],f5[x,y,z],f6[x,y,z],f7[x,y,z],f8[x,y,z],f9[x,y,z],f10[x,y,z],f11[x,y,z],f12[x,y,z],f13[x,y,z],f14[x,y,z],f15[x,y,z],f16[x,y,z],f17[x,y,z],f18[x,y,z],f19[x,y,z],f20[x,y,z],f21[x,y,z],f22[x,y,z],f23[x,y,z],f24[x,y,z],f25[x,y,z],f26[x,y,z],f27[x,y,z],f28[x,y,z],f29[x,y,z],f30[x,y,z],f31[x,y,z],f32[x,y,z],f33[x,y,z],f34[x,y,z],f35[x,y,z],f36[x,y,z],f37[x,y,z],f38[x,y,z],f39[x,y,z],f40[x,y,z],f41[x,y,z],f42[x,y,z],f43[x,y,z],f44[x,y,z],f45[x,y,z],f46[x,y,z],f47[x,y,z],f48[x,y,z],f49[x,y,z],f50[x,y,z],f51[x,y,z],f52[x,y,z],f53[x,y,z],f54[x,y,z],f55[x,y,z],f56[x,y,z],f57[x,y,z],f58[x,y,z],f59[x,y,z],f60[x,y,z],f61[x,y,z],f62[x,y,z],f63[x,y,z],f64[x,y,z],f65[x,y,z],f66[x,y,z]} fx1[x_,y_,z_]:=10x^9 fx2[x_,y_,z_]:=9x^8y fx3[x_,y_,z_]:=9x^8z fx4[x_,y_,z_]:=8x^7y^2 fx5[x_,y_,z_]:=8x^7y z fx6[x_,y_,z_]:=8x^7z^2 fx7[x_,y_,z_]:=7x^6y^3 fx8[x_,y_,z_]:=7x^6y^2z fx9[x_,y_,z_]:=7x^6y z^2 fx10[x_,y_,z_]:=7x^6z^3 fx11[x_,y_,z_]:=6x^5y^4 fx12[x_,y_,z_]:=6x^5y^3z fx13[x_,y_,z_]:=6x^5y^2z^2 fx14[x_,y_,z_]:=6x^5y z^3 fx15[x_,y_,z_]:=6x^5z^4 fx16[x_,y_,z_]:=5x^4y^5 fx17[x_,y_,z_]:=5x^4y^4z fx18[x_,y_,z_]:=5x^4y^3z^2 fx19[x_,y_,z_]:=5x^4y^2z^3 fx20[x_,y_,z_]:=5x^4y z^4 fx21[x_,y_,z_]:=5x^4z^5 fx22[x_,y_,z_]:=4x^3y^6 fx23[x_,y_,z_]:=4x^3y^5z fx24[x_,y_,z_]:=4x^3y^4z^2 fx25[x_,y_,z_]:=4x^3y^3z^3 fx26[x_,y_,z_]:=4x^3y^2z^4 fx27[x_,y_,z_]:=4x^3y z^5 fx28[x_,y_,z_]:=4x^3z^6 fx29[x_,y_,z_]:=3x^2y^7 fx30[x_,y_,z_]:=3x^2y^6z fx31[x_,y_,z_]:=3x^2y^5z^2 fx32[x_,y_,z_]:=3x^2y^4z^3 fx33[x_,y_,z_]:=3x^2y^3z^4 fx34[x_,y_,z_]:=3x^2y^2z^5 fx35[x_,y_,z_]:=3x^2y z^6 fx36[x_,y_,z_]:=3x^2z^7 fx37[x_,y_,z_]:=2x y^8 fx38[x_,y_,z_]:=2x y^7z fx39[x_,y_,z_]:=2x y^6z^2 fx40[x_,y_,z_]:=2x y^5z^3 fx41[x_,y_,z_]:=2x y^4z^4 fx42[x_,y_,z_]:=2x y^3z^5 fx43[x_,y_,z_]:=2x y^2z^6 fx44[x_,y_,z_]:=2x y z^7 fx45[x_,y_,z_]:=2x z^8 fx46[x_,y_,z_]:=y^9 fx47[x_,y_,z_]:=y^8z fx48[x_,y_,z_]:=y^7z^2 fx49[x_,y_,z_]:=y^6z^3 fx50[x_,y_,z_]:=y^5z^4 fx51[x_,y_,z_]:=y^4z^5 fx52[x_,y_,z_]:=y^3z^6 fx53[x_,y_,z_]:=y^2z^7 fx54[x_,y_,z_]:=y z^8 fx55[x_,y_,z_]:=z^9 fx56[x_,y_,z_]:=0 fx57[x_,y_,z_]:=0 fx58[x_,y_,z_]:=0 fx59[x_,y_,z_]:=0 fx60[x_,y_,z_]:=0 fx61[x_,y_,z_]:=0 fx62[x_,y_,z_]:=0 fx63[x_,y_,z_]:=0 fx64[x_,y_,z_]:=0 fx65[x_,y_,z_]:=0 fx66[x_,y_,z_]:=0 fx[x_,y_,z_]:={fx1[x,y,z],fx2[x,y,z],fx3[x,y,z],fx4[x,y,z],fx5[x,y,z],fx6[x,y,z],fx7[x,y,z],fx8[x,y,z],fx9[x,y,z],fx10[x,y,z],fx11[x,y,z],fx12[x,y,z],fx13[x,y,z],fx14[x,y,z],fx15[x,y,z],fx16[x,y,z],fx17[x,y,z],fx18[x,y,z],fx19[x,y,z],fx20[x,y,z],fx21[x,y,z],fx22[x,y,z],fx23[x,y,z],fx24[x,y,z],fx25[x,y,z],fx26[x,y,z],fx27[x,y,z],fx28[x,y,z],fx29[x,y,z],fx30[x,y,z],fx31[x,y,z],fx32[x,y,z],fx33[x,y,z],fx34[x,y,z],fx35[x,y,z],fx36[x,y,z],fx37[x,y,z],fx38[x,y,z],fx39[x,y,z],fx40[x,y,z],fx41[x,y,z],fx42[x,y,z],fx43[x,y,z],fx44[x,y,z],fx45[x,y,z],fx46[x,y,z],fx47[x,y,z],fx48[x,y,z],fx49[x,y,z],fx50[x,y,z],fx51[x,y,z],fx52[x,y,z],fx53[x,y,z],fx54[x,y,z],fx55[x,y,z],fx56[x,y,z],fx57[x,y,z],fx58[x,y,z],fx59[x,y,z],fx60[x,y,z],fx61[x,y,z],fx62[x,y,z],fx63[x,y,z],fx64[x,y,z],fx65[x,y,z],fx66[x,y,z]} fy1[x_,y_,z_]:=0 fy2[x_,y_,z_]:=x^9 fy3[x_,y_,z_]:=0 fy4[x_,y_,z_]:=2x^8y fy5[x_,y_,z_]:=x^8z fy6[x_,y_,z_]:=0 fy7[x_,y_,z_]:=3x^7y^2 fy8[x_,y_,z_]:=2x^7y z fy9[x_,y_,z_]:=x^7z^2 fy10[x_,y_,z_]:=0 fy11[x_,y_,z_]:=4x^6y^3 fy12[x_,y_,z_]:=3x^6y^2z fy13[x_,y_,z_]:=2x^6y z^2 fy14[x_,y_,z_]:=x^6z^3 fy15[x_,y_,z_]:=0 fy16[x_,y_,z_]:=5x^5y^4 fy17[x_,y_,z_]:=4x^5y^3z fy18[x_,y_,z_]:=3x^5y^2z^2 fy19[x_,y_,z_]:=2x^5y z^3 fy20[x_,y_,z_]:=x^5z^4 fy21[x_,y_,z_]:=0 fy22[x_,y_,z_]:=6x^4y^5 fy23[x_,y_,z_]:=5x^4y^4z fy24[x_,y_,z_]:=4x^4y^3z^2 fy25[x_,y_,z_]:=3x^4y^2z^3 fy26[x_,y_,z_]:=2x^4y z^4 fy27[x_,y_,z_]:=x^4z^5 fy28[x_,y_,z_]:=0 fy29[x_,y_,z_]:=7x^3y^6 fy30[x_,y_,z_]:=6x^3y^5z fy31[x_,y_,z_]:=5x^3y^4z^2 fy32[x_,y_,z_]:=4x^3y^3z^3 fy33[x_,y_,z_]:=3x^3y^2z^4 fy34[x_,y_,z_]:=2x^3y z^5 fy35[x_,y_,z_]:=x^3z^6 fy36[x_,y_,z_]:=0 fy37[x_,y_,z_]:=8x^2y^7 fy38[x_,y_,z_]:=7x^2y^6z fy39[x_,y_,z_]:=6x^2y^5z^2 fy40[x_,y_,z_]:=5x^2y^4z^3 fy41[x_,y_,z_]:=4x^2y^3z^4 fy42[x_,y_,z_]:=3x^2y^2z^5 fy43[x_,y_,z_]:=2x^2y z^6 fy44[x_,y_,z_]:=x^2z^7 fy45[x_,y_,z_]:=0 fy46[x_,y_,z_]:=9x y^8 fy47[x_,y_,z_]:=8x y^7z fy48[x_,y_,z_]:=7x y^6z^2 fy49[x_,y_,z_]:=6x y^5z^3 fy50[x_,y_,z_]:=5x y^4z^4 fy51[x_,y_,z_]:=4x y^3z^5 fy52[x_,y_,z_]:=3x y^2z^6 fy53[x_,y_,z_]:=2x y z^7 fy54[x_,y_,z_]:=x z^8 fy55[x_,y_,z_]:=0 fy56[x_,y_,z_]:=10y^9 fy57[x_,y_,z_]:=9y^8z fy58[x_,y_,z_]:=8y^7z^2 fy59[x_,y_,z_]:=7y^6z^3 fy60[x_,y_,z_]:=6y^5z^4 fy61[x_,y_,z_]:=5y^4z^5 fy62[x_,y_,z_]:=4y^3z^6 fy63[x_,y_,z_]:=3y^2z^7 fy64[x_,y_,z_]:=2y z^8 fy65[x_,y_,z_]:=z^9 fy66[x_,y_,z_]:=0 fy[x_,y_,z_]:={fy1[x,y,z],fy2[x,y,z],fy3[x,y,z],fy4[x,y,z],fy5[x,y,z],fy6[x,y,z],fy7[x,y,z],fy8[x,y,z],fy9[x,y,z],fy10[x,y,z],fy11[x,y,z],fy12[x,y,z],fy13[x,y,z],fy14[x,y,z],fy15[x,y,z],fy16[x,y,z],fy17[x,y,z],fy18[x,y,z],fy19[x,y,z],fy20[x,y,z],fy21[x,y,z],fy22[x,y,z],fy23[x,y,z],fy24[x,y,z],fy25[x,y,z],fy26[x,y,z],fy27[x,y,z],fy28[x,y,z],fy29[x,y,z],fy30[x,y,z],fy31[x,y,z],fy32[x,y,z],fy33[x,y,z],fy34[x,y,z],fy35[x,y,z],fy36[x,y,z],fy37[x,y,z],fy38[x,y,z],fy39[x,y,z],fy40[x,y,z],fy41[x,y,z],fy42[x,y,z],fy43[x,y,z],fy44[x,y,z],fy45[x,y,z],fy46[x,y,z],fy47[x,y,z],fy48[x,y,z],fy49[x,y,z],fy50[x,y,z],fy51[x,y,z],fy52[x,y,z],fy53[x,y,z],fy54[x,y,z],fy55[x,y,z],fy56[x,y,z],fy57[x,y,z],fy58[x,y,z],fy59[x,y,z],fy60[x,y,z],fy61[x,y,z],fy62[x,y,z],fy63[x,y,z],fy64[x,y,z],fy65[x,y,z],fy66[x,y,z]} fz1[x_,y_,z_]:=0 fz2[x_,y_,z_]:=0 fz3[x_,y_,z_]:=x^9 fz4[x_,y_,z_]:=0 fz5[x_,y_,z_]:=x^8y fz6[x_,y_,z_]:=2x^8z fz7[x_,y_,z_]:=0 fz8[x_,y_,z_]:=x^7y^2 fz9[x_,y_,z_]:=2x^7y z fz10[x_,y_,z_]:=3x^7z^2 fz11[x_,y_,z_]:=0 fz12[x_,y_,z_]:=x^6y^3 fz13[x_,y_,z_]:=2x^6y^2z fz14[x_,y_,z_]:=3x^6y z^2 fz15[x_,y_,z_]:=4x^6z^3 fz16[x_,y_,z_]:=0 fz17[x_,y_,z_]:=x^5y^4 fz18[x_,y_,z_]:=2x^5y^3z fz19[x_,y_,z_]:=3x^5y^2z^2 fz20[x_,y_,z_]:=4x^5y z^3 fz21[x_,y_,z_]:=5x^5z^4 fz22[x_,y_,z_]:=0 fz23[x_,y_,z_]:=x^4y^5 fz24[x_,y_,z_]:=2x^4y^4z fz25[x_,y_,z_]:=3x^4y^3z^2 fz26[x_,y_,z_]:=4x^4y^2z^3 fz27[x_,y_,z_]:=5x^4y z^4 fz28[x_,y_,z_]:=6x^4z^5 fz29[x_,y_,z_]:=0 fz30[x_,y_,z_]:=x^3y^6 fz31[x_,y_,z_]:=2x^3y^5z fz32[x_,y_,z_]:=3x^3y^4z^2 fz33[x_,y_,z_]:=4x^3y^3z^3 fz34[x_,y_,z_]:=5x^3y^2z^4 fz35[x_,y_,z_]:=6x^3y z^5 fz36[x_,y_,z_]:=7x^3z^6 fz37[x_,y_,z_]:=0 fz38[x_,y_,z_]:=x^2y^7 fz39[x_,y_,z_]:=2x^2y^6z fz40[x_,y_,z_]:=3x^2y^5z^2 fz41[x_,y_,z_]:=4x^2y^4z^3 fz42[x_,y_,z_]:=5x^2y^3z^4 fz43[x_,y_,z_]:=6x^2y^2z^5 fz44[x_,y_,z_]:=7x^2y z^6 fz45[x_,y_,z_]:=8x^2z^7 fz46[x_,y_,z_]:=0 fz47[x_,y_,z_]:=x y^8 fz48[x_,y_,z_]:=2x y^7z fz49[x_,y_,z_]:=3x y^6z^2 fz50[x_,y_,z_]:=4x y^5z^3 fz51[x_,y_,z_]:=5x y^4z^4 fz52[x_,y_,z_]:=6x y^3z^5 fz53[x_,y_,z_]:=7x y^2z^6 fz54[x_,y_,z_]:=8x y z^7 fz55[x_,y_,z_]:=9x z^8 fz56[x_,y_,z_]:=0 fz57[x_,y_,z_]:=y^9 fz58[x_,y_,z_]:=2y^8z fz59[x_,y_,z_]:=3y^7z^2 fz60[x_,y_,z_]:=4y^6z^3 fz61[x_,y_,z_]:=5y^5z^4 fz62[x_,y_,z_]:=6y^4z^5 fz63[x_,y_,z_]:=7y^3z^6 fz64[x_,y_,z_]:=8y^2z^7 fz65[x_,y_,z_]:=9y z^8 fz66[x_,y_,z_]:=10z^9 fz[x_,y_,z_]:={fz1[x,y,z],fz2[x,y,z],fz3[x,y,z],fz4[x,y,z],fz5[x,y,z],fz6[x,y,z],fz7[x,y,z],fz8[x,y,z],fz9[x,y,z],fz10[x,y,z],fz11[x,y,z],fz12[x,y,z],fz13[x,y,z],fz14[x,y,z],fz15[x,y,z],fz16[x,y,z],fz17[x,y,z],fz18[x,y,z],fz19[x,y,z],fz20[x,y,z],fz21[x,y,z],fz22[x,y,z],fz23[x,y,z],fz24[x,y,z],fz25[x,y,z],fz26[x,y,z],fz27[x,y,z],fz28[x,y,z],fz29[x,y,z],fz30[x,y,z],fz31[x,y,z],fz32[x,y,z],fz33[x,y,z],fz34[x,y,z],fz35[x,y,z],fz36[x,y,z],fz37[x,y,z],fz38[x,y,z],fz39[x,y,z],fz40[x,y,z],fz41[x,y,z],fz42[x,y,z],fz43[x,y,z],fz44[x,y,z],fz45[x,y,z],fz46[x,y,z],fz47[x,y,z],fz48[x,y,z],fz49[x,y,z],fz50[x,y,z],fz51[x,y,z],fz52[x,y,z],fz53[x,y,z],fz54[x,y,z],fz55[x,y,z],fz56[x,y,z],fz57[x,y,z],fz58[x,y,z],fz59[x,y,z],fz60[x,y,z],fz61[x,y,z],fz62[x,y,z],fz63[x,y,z],fz64[x,y,z],fz65[x,y,z],fz66[x,y,z]} fd66[x_,y_,z_]:={f1[x,y,z],f2[x,y,z],f3[x,y,z],f4[x,y,z],f5[x,y,z],f6[x,y,z],f7[x,y,z],f8[x,y,z],f9[x,y,z],f10[x,y,z],f11[x,y,z],f12[x,y,z],f13[x,y,z],f14[x,y,z],f15[x,y,z],f16[x,y,z],f17[x,y,z],f18[x,y,z],f19[x,y,z],f20[x,y,z],f21[x,y,z],f22[x,y,z],f23[x,y,z],f24[x,y,z],f25[x,y,z],f26[x,y,z],f27[x,y,z],f28[x,y,z],f29[x,y,z],f30[x,y,z],f31[x,y,z],f32[x,y,z],f33[x,y,z],f34[x,y,z],f35[x,y,z],f36[x,y,z],f37[x,y,z],f38[x,y,z],f39[x,y,z],f40[x,y,z],f41[x,y,z],f42[x,y,z],f43[x,y,z],f44[x,y,z],f45[x,y,z],f46[x,y,z],f47[x,y,z],f48[x,y,z],f49[x,y,z],f50[x,y,z],f51[x,y,z],f52[x,y,z],f53[x,y,z],f54[x,y,z],f55[x,y,z],f56[x,y,z],f57[x,y,z],f58[x,y,z],f59[x,y,z],f60[x,y,z],f61[x,y,z],f62[x,y,z],f63[x,y,z],f64[x,y,z],f65[x,y,z]} fxd66[x_,y_,z_]:={fx1[x,y,z],fx2[x,y,z],fx3[x,y,z],fx4[x,y,z],fx5[x,y,z],fx6[x,y,z],fx7[x,y,z],fx8[x,y,z],fx9[x,y,z],fx10[x,y,z],fx11[x,y,z],fx12[x,y,z],fx13[x,y,z],fx14[x,y,z],fx15[x,y,z],fx16[x,y,z],fx17[x,y,z],fx18[x,y,z],fx19[x,y,z],fx20[x,y,z],fx21[x,y,z],fx22[x,y,z],fx23[x,y,z],fx24[x,y,z],fx25[x,y,z],fx26[x,y,z],fx27[x,y,z],fx28[x,y,z],fx29[x,y,z],fx30[x,y,z],fx31[x,y,z],fx32[x,y,z],fx33[x,y,z],fx34[x,y,z],fx35[x,y,z],fx36[x,y,z],fx37[x,y,z],fx38[x,y,z],fx39[x,y,z],fx40[x,y,z],fx41[x,y,z],fx42[x,y,z],fx43[x,y,z],fx44[x,y,z],fx45[x,y,z],fx46[x,y,z],fx47[x,y,z],fx48[x,y,z],fx49[x,y,z],fx50[x,y,z],fx51[x,y,z],fx52[x,y,z],fx53[x,y,z],fx54[x,y,z],fx55[x,y,z],fx56[x,y,z],fx57[x,y,z],fx58[x,y,z],fx59[x,y,z],fx60[x,y,z],fx61[x,y,z],fx62[x,y,z],fx63[x,y,z],fx64[x,y,z],fx65[x,y,z]} fyd66[x_,y_,z_]:={fy1[x,y,z],fy2[x,y,z],fy3[x,y,z],fy4[x,y,z],fy5[x,y,z],fy6[x,y,z],fy7[x,y,z],fy8[x,y,z],fy9[x,y,z],fy10[x,y,z],fy11[x,y,z],fy12[x,y,z],fy13[x,y,z],fy14[x,y,z],fy15[x,y,z],fy16[x,y,z],fy17[x,y,z],fy18[x,y,z],fy19[x,y,z],fy20[x,y,z],fy21[x,y,z],fy22[x,y,z],fy23[x,y,z],fy24[x,y,z],fy25[x,y,z],fy26[x,y,z],fy27[x,y,z],fy28[x,y,z],fy29[x,y,z],fy30[x,y,z],fy31[x,y,z],fy32[x,y,z],fy33[x,y,z],fy34[x,y,z],fy35[x,y,z],fy36[x,y,z],fy37[x,y,z],fy38[x,y,z],fy39[x,y,z],fy40[x,y,z],fy41[x,y,z],fy42[x,y,z],fy43[x,y,z],fy44[x,y,z],fy45[x,y,z],fy46[x,y,z],fy47[x,y,z],fy48[x,y,z],fy49[x,y,z],fy50[x,y,z],fy51[x,y,z],fy52[x,y,z],fy53[x,y,z],fy54[x,y,z],fy55[x,y,z],fy56[x,y,z],fy57[x,y,z],fy58[x,y,z],fy59[x,y,z],fy60[x,y,z],fy61[x,y,z],fy62[x,y,z],fy63[x,y,z],fy64[x,y,z],fy65[x,y,z]} BetuBd66[p_,q_]:={ fd66[p,1,1], fxd66[p,1,1], fd66[-p,1,1], fxd66[-p,1,1], fd66[p,-1,1], fxd66[p,-1,1], fyd66[p,-1,1], fd66[p,1,-1], fxd66[p,1,-1], fyd66[p,1,-1], fd66[1,p,1], fxd66[1,p,1], fyd66[1,p,1], fd66[-1,p,1], fxd66[-1,p,1], fyd66[-1,p,1], fd66[1,-p,1], fxd66[1,-p,1], fyd66[1,-p,1], fd66[1,p,-1], fxd66[1,p,-1], fd66[1,1,p], fxd66[1,1,p], fyd66[1,1,p], fd66[-1,1,p], fxd66[-1,1,p], fyd66[-1,1,p], fd66[1,-1,p], fxd66[1,-1,p], fyd66[1,-1,p], fd66[1,1,-p], fxd66[1,1,-p], fyd66[1,1,-p], fd66[q,1,0], fxd66[q,1,0], fd66[q,0,1], fxd66[q,0,1], fyd66[q,0,1], fd66[-q,1,0], fyd66[-q,1,0], fd66[-q,0,1], fxd66[-q,0,1], fyd66[-q,0,1], fd66[1,q,0], fxd66[1,q,0], fd66[1,0,q], fxd66[1,0,q], fyd66[1,0,q], fd66[1,0,-q], fxd66[1,0,-q], fyd66[1,0,-q], fd66[0,q,1], fxd66[0,q,1], fyd66[0,q,1], fd66[0,-q,1], fxd66[0,-q,1], fyd66[0,-q,1], fd66[0,1,q], fxd66[0,1,q], fyd66[0,1,q], fd66[0,1,-q], fxd66[0,1,-q], fyd66[0,1,-q], fd66[1,-q,0], fxd66[1,-q,0]} MatrixRank[BetuBd66[3,2]] (* = 65 *) Factor[Det[BetuBd66[p,q]]] (* = 2417851639229258349412352 p^45 (p-1)^38 (p+1)^38 (p^2+2)^5 q^61 (q-1)^28 (q+1)^28 (q^2+1)^28 (q^2-p^2-1)^3 (-1 + q^2 + p^2 q^2)^3 (1 - 2 q - p^2 q + q^2)^5 (1 + 2 q + p^2 q + q^2)^5 (1 + 2 p^2 - 2 q^2 - p^4 q^2 + q^4 + 2 p^2 q^4)^9 *) (*================ $\fre_{t,u}^B \in \cE(\cP_{3,5}) ========================*) f1[x_,y_,z_]:=x^5 f2[x_,y_,z_]:=x^4y f3[x_,y_,z_]:=x^4z f4[x_,y_,z_]:=x^3y^2 f5[x_,y_,z_]:=x^3y z f6[x_,y_,z_]:=x^3z^2 f7[x_,y_,z_]:=x^2y^3 f8[x_,y_,z_]:=x^2y^2z f9[x_,y_,z_]:=x^2y z^2 f10[x_,y_,z_]:=x^2z^3 f11[x_,y_,z_]:=x y^4 f12[x_,y_,z_]:=x y^3z f13[x_,y_,z_]:=x y^2z^2 f14[x_,y_,z_]:=x y z^3 f15[x_,y_,z_]:=x z^4 f16[x_,y_,z_]:=y^5 f17[x_,y_,z_]:=y^4z f18[x_,y_,z_]:=y^3z^2 f19[x_,y_,z_]:=y^2z^3 f20[x_,y_,z_]:=y z^4 f21[x_,y_,z_]:=z^5 f[x_,y_,z_] := {f1[x,y,z],f2[x,y,z],f3[x,y,z],f4[x,y,z],f5[x,y,z],f6[x,y,z],f7[x,y,z],f8[x,y,z],f9[x,y,z],f10[x,y,z],f11[x,y,z],f12[x,y,z],f13[x,y,z],f14[x,y,z],f15[x,y,z],f16[x,y,z],f17[x,y,z],f18[x,y,z],f19[x,y,z],f20[x,y,z],f21[x,y,z]} fx1[x_,y_,z_]:=5x^4 fx2[x_,y_,z_]:=4x^3y fx3[x_,y_,z_]:=4x^3z fx4[x_,y_,z_]:=3x^2y^2 fx5[x_,y_,z_]:=3x^2y z fx6[x_,y_,z_]:=3x^2z^2 fx7[x_,y_,z_]:=2x y^3 fx8[x_,y_,z_]:=2x y^2z fx9[x_,y_,z_]:=2x y z^2 fx10[x_,y_,z_]:=2x z^3 fx11[x_,y_,z_]:=y^4 fx12[x_,y_,z_]:=y^3z fx13[x_,y_,z_]:=y^2z^2 fx14[x_,y_,z_]:=y z^3 fx15[x_,y_,z_]:=z^4 fx16[x_,y_,z_]:=0 fx17[x_,y_,z_]:=0 fx18[x_,y_,z_]:=0 fx19[x_,y_,z_]:=0 fx20[x_,y_,z_]:=0 fx21[x_,y_,z_]:=0 fx[x_,y_,z_] := {fx1[x,y,z],fx2[x,y,z],fx3[x,y,z],fx4[x,y,z],fx5[x,y,z],fx6[x,y,z],fx7[x,y,z],fx8[x,y,z],fx9[x,y,z],fx10[x,y,z],fx11[x,y,z],fx12[x,y,z],fx13[x,y,z],fx14[x,y,z],fx15[x,y,z],fx16[x,y,z],fx17[x,y,z],fx18[x,y,z],fx19[x,y,z],fx20[x,y,z],fx21[x,y,z]} fy1[x_,y_,z_]:=0 fy2[x_,y_,z_]:=x^4 fy3[x_,y_,z_]:=0 fy4[x_,y_,z_]:=2x^3y fy5[x_,y_,z_]:=x^3z fy6[x_,y_,z_]:=0 fy7[x_,y_,z_]:=3x^2y^2 fy8[x_,y_,z_]:=2x^2y z fy9[x_,y_,z_]:=x^2z^2 fy10[x_,y_,z_]:=0 fy11[x_,y_,z_]:=4x y^3 fy12[x_,y_,z_]:=3x y^2z fy13[x_,y_,z_]:=2x y z^2 fy14[x_,y_,z_]:=x z^3 fy15[x_,y_,z_]:=0 fy16[x_,y_,z_]:=5y^4 fy17[x_,y_,z_]:=4y^3z fy18[x_,y_,z_]:=3y^2z^2 fy19[x_,y_,z_]:=2y z^3 fy20[x_,y_,z_]:=z^4 fy21[x_,y_,z_]:=0 fy[x_,y_,z_] := {fy1[x,y,z],fy2[x,y,z],fy3[x,y,z],fy4[x,y,z],fy5[x,y,z],fy6[x,y,z],fy7[x,y,z],fy8[x,y,z],fy9[x,y,z],fy10[x,y,z],fy11[x,y,z],fy12[x,y,z],fy13[x,y,z],fy14[x,y,z],fy15[x,y,z],fy16[x,y,z],fy17[x,y,z],fy18[x,y,z],fy19[x,y,z],fy20[x,y,z],fy21[x,y,z]} fz1[x_,y_,z_]:=0 fz2[x_,y_,z_]:=0 fz3[x_,y_,z_]:=x^4 fz4[x_,y_,z_]:=0 fz5[x_,y_,z_]:=x^3y fz6[x_,y_,z_]:=2x^3z fz7[x_,y_,z_]:=0 fz8[x_,y_,z_]:=x^2y^2 fz9[x_,y_,z_]:=2x^2y z fz10[x_,y_,z_]:=3x^2z^2 fz11[x_,y_,z_]:=0 fz12[x_,y_,z_]:=x y^3 fz13[x_,y_,z_]:=2x y^2z fz14[x_,y_,z_]:=3x y z^2 fz15[x_,y_,z_]:=4x z^3 fz16[x_,y_,z_]:=0 fz17[x_,y_,z_]:=y^4 fz18[x_,y_,z_]:=2y^3z fz19[x_,y_,z_]:=3y^2z^2 fz20[x_,y_,z_]:=4y z^3 fz21[x_,y_,z_]:=5z^4 fz[x_,y_,z_] := {fz1[x,y,z],fz2[x,y,z],fz3[x,y,z],fz4[x,y,z],fz5[x,y,z],fz6[x,y,z],fz7[x,y,z],fz8[x,y,z],fz9[x,y,z],fz10[x,y,z],fz11[x,y,z],fz12[x,y,z],fz13[x,y,z],fz14[x,y,z],fz15[x,y,z],fz16[x,y,z],fz17[x,y,z],fz18[x,y,z],fz19[x,y,z],fz20[x,y,z],fz21[x,y,z]} fd21[x_,y_,z_] := {f1[x,y,z],f2[x,y,z],f3[x,y,z],f4[x,y,z],f5[x,y,z],f6[x,y,z],f7[x,y,z],f8[x,y,z],f9[x,y,z],f10[x,y,z],f11[x,y,z],f12[x,y,z],f13[x,y,z],f14[x,y,z],f15[x,y,z],f16[x,y,z],f17[x,y,z],f18[x,y,z],f19[x,y,z],f20[x,y,z]} fxd21[x_,y_,z_] := {fx1[x,y,z],fx2[x,y,z],fx3[x,y,z],fx4[x,y,z],fx5[x,y,z],fx6[x,y,z],fx7[x,y,z],fx8[x,y,z],fx9[x,y,z],fx10[x,y,z],fx11[x,y,z],fx12[x,y,z],fx13[x,y,z],fx14[x,y,z],fx15[x,y,z],fx16[x,y,z],fx17[x,y,z],fx18[x,y,z],fx19[x,y,z],fx20[x,y,z]} fyd21[x_,y_,z_] := {fy1[x,y,z],fy2[x,y,z],fy3[x,y,z],fy4[x,y,z],fy5[x,y,z],fy6[x,y,z],fy7[x,y,z],fy8[x,y,z],fy9[x,y,z],fy10[x,y,z],fy11[x,y,z],fy12[x,y,z],fy13[x,y,z],fy14[x,y,z],fy15[x,y,z],fy16[x,y,z],fy17[x,y,z],fy18[x,y,z],fy19[x,y,z],fy20[x,y,z]} fzd21[x_,y_,z_] := {fz1[x,y,z],fz2[x,y,z],fz3[x,y,z],fz4[x,y,z],fz5[x,y,z],fz6[x,y,z],fz7[x,y,z],fz8[x,y,z],fz9[x,y,z],fz10[x,y,z],fz11[x,y,z],fz12[x,y,z],fz13[x,y,z],fz14[x,y,z],fz15[x,y,z],fz16[x,y,z],fz17[x,y,z],fz18[x,y,z],fz19[x,y,z],fz20[x,y,z]} AetuB[t_,u_]:={ f[t,1,1], fx[t,1,1], fy[t,1,1], f[1,t,1], fx[1,t,1], fy[1,t,1], f[1,1,t], fx[1,1,t], fy[1,1,t], f[0,u,1], fy[0,u,1], f[0,1,u], fz[0,1,u], f[u,0,1], fx[u,0,1], f[1,0,u], fz[1,0,u], f[u,1,0], fx[u,1,0], f[1,u,0], fy[1,u,0]} Factor[Det[AetuB[t,u]]] (* = 0 *) AetuBd21[t_,u_]:={ fd21[t,1,1], fxd21[t,1,1], fyd21[t,1,1], fd21[1,t,1], fxd21[1,t,1], fyd21[1,t,1], fd21[1,1,t], fxd21[1,1,t], fyd21[1,1,t], fd21[0,u,1], fyd21[0,u,1], fd21[0,1,u], fzd21[0,1,u], fd21[u,0,1], fxd21[u,0,1], fd21[1,0,u], fzd21[1,0,u], fd21[u,1,0], fxd21[u,1,0], fd21[1,u,0]} Factor[Det[AetuBd21[t,u]]] (* = t^5 (t-1)^11 (t+2) u^7 (u-1)^10 (u+1)^11 (1 - 2 u - 4 t u - t^2 u + u^2)^2 (1 + 2 t - 2 u - t^2 u + u^2 + 2 t u^2)^3 *) (*============================ Section 4.3.3 ================================*) (*============================ Section 4.3.4 ================================*)