(* This file contains tools for Mathematica to check my article. *) (* Theorem 3.1. *) (* Definitions of functions *) 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 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 s0[a_,b_,c_] := S3[a,b,c]-3U[a,b,c] s1[a_,b_,c_] := S21[a,b,c]-3U[a,b,c] s2[a_,b_,c_] := S12[a,b,c]-3U[a,b,c] s3[a_,b_,c_] := U[a,b,c] Solve[{S3[0, s, 1] + p S21[0, s, 1] + q S12[0, s, 1] == 0, 3 s^2 + 2 s p + q == 0}, {p, q}] (* 3s^2 = D[S3[0,s,1],s], 2s = D[S21[0,s,1],s], 1 = D[S12[0,s,1],s] *) (* Solution: p = (1-2s^3)/s^2), q = (s^3-2)/s. Thus let *) FrakF[a_,b_,c_,s_] := s^2 S3[a,b,c] - (2s^3-1)S21[a,b,c] + (s^4-2s) S12[a,b,c] - 3(s^4-2s^3+s^2-2s+1)U[a,b,c] Disc3p[p_,q_] := 4p^3 + 4q^3 + 27 - p^2q^2 - 18p q (* Check ${\frak f}_s(a,b,c) \geq 0$, *) Factor[FrakF[a,1-k(1-a),1,s] - (1-a)^2 (a(1- k s)^2(k+s^2) + (1+(1-k)s^2) (1-k-s)^2)] (* Check ${\frak f}_s(0,s,1) = 0$, *) Factor[FrakF[0,s,1,s]] (* Check $\disc_3^+$ is a discriminant. *) Factor[Disc3p[-(2s^3-1)/s^2, (s^4-2s)/s^2]] (* Defining equation of $X_3^c$ *) F3[w_,x_,y_,z_] := x^3 + y^3 + 9z^3 - 6 x y z - x y w + 3z^2 w + z w^2 G3[w_,x_,y_,z_] := x^3 - w x y + y^3 + w^2 z - 3 w x z + 9 x^2 z - 3 w y z - 9 x y z + 9 y^2 z Factor[F3[S3[a,b,c],S21[a,b,c],S12[a,b,c],U[a,b,c]]] (* = 0 *) Factor[G3[s0[a,b,c],s1[a,b,c],s2[a,b,c],s3[a,b,c]]] (* = 0 *) F3w[w_,x_,y_,z_] := -x y + 2 w z + 3 z^2 (* = D[F3[w,x,y,z],w] *) F3x[w_,x_,y_,z_] := 3 x^2 - w y - 6 y z (* = D[F3[w,x,y,z],x] *) F3y[w_,x_,y_,z_] := -w x + 3 y^2 - 6 x z (* = D[F3[w,x,y,z],y] *) F3z[w_,x_,y_,z_] := w^2 - 6 x y + 6 w z + 27 z^2 (* = D[F3[w,x,y,z],z] *) Eliminate[{p0 w + p1 x + p2 y + p3 z ==0, p0 == -x y + 2 w z + 3 z^2, p1 == 3 x^2 - w y - 6 y z, p2 == -w x + 3 y^2 - 6 x z, p3 == w^2 - 6 x y + 6 w z + 27 z^2}, {w,x,y,z}] (* Result of elimination: Defining equation of main component. It is empty. *) disc0[p0_,p1_,p2_,p3_]:= 9 p0^3 + p1^3 - 3 p0 p1 p2 + p2^3 - 3 p0^2 p3 - p1 p2 p3 + p0 p3^2 ContourPlot3D[disc0[1, p, q, r] == 0, {p, -10, 10}, {q, -10, 10}, {r, -10, 10}] (* Elimination $\displaystyle \sum_{i=0}^N p_i f_i(P(s_1,\ldots, s_r)) = 0$ and $$\sum_{i=0}^N p_i {\partial f_i \over \partial s_j}(P(s_1,\ldots, s_r)) = 0$$ *) Gw[p0_,p1_,p2_,p3_,s_,t_]:=p0 S3[s,t,1] + p1 S21[s,t,1] + p2 S12[s,t,1] + p3 U[s,t,1] (* D[Gw[p0,p1,p2,p3,s,t],s] *) Gws[p0_,p1_,p2_,p3_,s_,t_] := p1 + 2 p2 s + 3 p0 s^2 + p3 t + 2 p1 s t + p2 t^2 (* D[Gw[p0,p1,p2,p3,s,t],t] *) Gwt[p0_,p1_,p2_,p3_,s_,t_] := p2 + p3 s + p1 s^2 + 2 p1 t + 2 p2 s t + 3 p0 t^2 Eliminate[{Gw[p0,p1,p2,p3,s,t]==0, p1 + 2 p2 s + 3 p0 s^2 + p3 t + 2 p1 s t + p2 t^2==0, p2 + p3 s + p1 s^2 + 2 p1 t + 2 p2 s t + 3 p0 t^2 ==0}, {s,t}] (* (3 p0 + 3 p1 + 3 p2 + p3) \times (9 p0^2 - 9 p0 p1 + 9 p1^2 - 9 p0 p2 - 9 p1 p2 + 9 p2^2 + 6 p0 p3 - 3 p1 p3 - 3 p2 p3 + p3^2) \times (9 p0^3 + p1^3 - 3 p0 p1 p2 + p2^3 - 3 p0^2 p3 - p1 p2 p3 + p0 p3^2) *) (* Edge Component *) Factor[D[Gw[p0, p1, p2, p3, 0, s], s]] (* = p2 + 2 p1 s + 3 p0 s^2 *) Eliminate[{Gw[p0,p1,p2,p3,0,s]==0, p2 + 2 p1 s + 3 p0 s^2 ==0}, s] (* = 27 p0^4 + 4 p0 p1^3 - 18 p0^2 p1 p2 - p1^2 p2^2 + 4 p0 p2^3 = p0^4 Disc3p[p1/p0, p2/p0] *) (* Anothe method to obtaon FrakF[a,b,c,s] as {\Cal P} \cap {\Cal H}_{0,s} *) Expand[ S3[0, s, 1] + p S21[0, s, 1] + q S12[0, s, 1] + r U[0, s, 1]] (* = 1 + q s + p s^2 + s^3 *) Eliminate[{Disc3p[p, q] == 0, 1 + q s + p s^2 + s^3 == 0}, q] (* = (4 + p^2 s + 2 p s^2 + s^3) (-1 + p s^2 + 2 s^3)^2. Thus p=(1-2s^3)/s^2 *) Factor[12 q s + 13 q^2 s^2 + (12 + 6 q^3) s^3 + q (6 + q^3) s^4 - 4 q^2 s^5 + (-15 - 2 q^3) s^6 - 6 q s^7 + q^2 s^8 + 4 s^9 + 4] (* = (4 + p^2 s + 2 p s^2 + s^3) (-1 + p s^2 + 2 s^3)^2 Thus q=(2s^3-1)/s^2 *) (*---------------------------------------------------------------------------*) (* Proposition 4.1 *) (* Definitions. Also use the definitions of Theprem 3.1. *) 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] T21[a_,b_,c_] := (a^2b+b^2c+c^2a)+(a b^2+b c^2+c a^2) s0[a_,b_,c_] := S4[a,b,c]-US1[a,b,c] s1[a_,b_,c_] := S31[a,b,c]-US1[a,b,c] s2[a_,b_,c_] := S13[a,b,c]-US1[a,b,c] s3[a_,b_,c_] := S22[a,b,c]-US1[a,b,c] s4[a_,b_,c_] := US1[a,b,c] (* Calculate the edge discriminant *) Eliminate[{p0 s0[0,s,1] + p1 s1[0,s,1] + p2 s2[0,s,1] + p3 s3[0,s,1] == 0, p0 (4s^3) + p1 (3s^2) + p2 + p3 (2s)==0}, s] Disc4cp[p0_,p1_,p2_,p3_]:= 256 p0^6 - 27 p0^2 p1^4 - 192 p0^4 p1 p2 - 6 p0^2 p1^2 p2^2 - 4 p1^3 p2^3 - 27 p0^2 p2^4 + 144 p0^3 p1^2 p3 + 18 p0 p1^3 p2 p3 + 144 p0^3 p2^2 p3 + 18 p0 p1 p2^3 p3 - 128 p0^4 p3^2 - 80 p0^2 p1 p2 p3^2 + p1^2 p2^2 p3^2 - 4 p0 p1^2 p3^3 - 4 p0 p2^2 p3^3 + 16 p0^2 p3^4 FrakP[s_,t_] := -(2 S31[s, t, 1] - S13[s, t, 1] - US1[s, t, 1])/(S22[s, t, 1] - US1[s, t, 1]) (* FrakP[s_,t_]:=1-(s+t+1)(T21[s,t,1]-6s t-3De[s,t,1])/(2(S22[s,t,1]- s t(s+t+1))) *) FrakQ[s_,t_] := -(2 S13[s, t, 1] - S31[s, t, 1] - US1[s, t, 1])/(S22[s, t, 1] - US1[s, t, 1]) (* FrakQ[s_,t_]:=1-(s+t+1)(T21[s,t,1]-6s t+3De[s,t,1])/(2(S22[s,t,1]- s t(s+t+1))) *) FrakG[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] 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] FrakK[a_,b_,c_,s_,t_] := s^2 S4[a,b,c] - (2s^3-s t)S31[a,b,c] + (s^3 t - 2 s) S13[a,b,c] + (s^4 - 2s^2t + 1) S22[a,b,c] + (s^2 - (s-1)^2 (s^2 + s t + 1)) US1[a,b,c] Disc40[p_,q_,r_] := 3(r+1) - (p^2 + p q + q^2) Disc4p[p_,q_,r_] := p^2 q^2 r^2 - 4 p^3 q^3 + 18 p^3 q r + 18 p q^3 r - 4 p^2 r^3 - 4 q^2 r^3 - 27 p^4 - 27 q^4 + 16 r^4- 6 p^2 q^2 - 80 p q r^2 + 144 p^2 r + 144 q^2 r - 192 p q - 128 r^2 + 256 (* Check $\displaystyle 6 {\frak g}_{p,q} = \sum_{\rm cyclic} (2a^2 - b^2 - c^2 + pab - (p+q)bc + qca)^2 \geq 0$. *) F41[a_,b_,c_]:=(2a^2-b^2-c^2+p a b-(p+q)b c+q c a)^2 Factor[6 FrakG[a,b,c,p,q] - (F41[a,b,c]+F41[b,c,a]+F41[c,a,b])] (* Check ${\frak g}_{{\frak p}(s,t),{\frak q}(s,t)}(s,t,1) = 0$ *) Factor[FrakG[s,t,1,FrakP[s,t],FrakQ[s,t]]] (* Check ${\frak g}_{p,q}(1,1,1) = 0$ *) Factor[FrakG[1,1,1,p,q]] (* Chack ${\frak h}_s(a,b,c) = (S_{1,3}-US_1)\left(s - {S_{2,2}-US_1 \over S_{1,3}-US_1}\right)^2 + {US_1(S_2-S_{1,1})^2 \over S_{1,3}-US_1} *) Factor[FrakH[a,b,c,s] - ((S13[a,b,c]-US1[a,b,c])( s - (S22[a,b,c]-US1[a,b,c])/(S13[a,b,c]-US1[a,b,c]))^2 + US1[a,b,c](S2[a,b,c]-S11[a,b,c])^2/(S13[a,b,c]-US1[a,b,c]))] (* Chack ${\frak k}_{s,t} = s^2 {\frak g}_{1/s - 2s, s - 2/s} + s(t-1) {\frak h}_s$ *) Factor[FrakK[a,b,c,s,t] - (s^2 FrakG[a,b,c,1/s-2s, s-2/s] + s(t-1) FrakH[a,b,c,s])] (* Chack ${\frak h}_s(0,s,1) = 0$ *) Factor[FrakH[0,s,1,s]] (*---------------------------------------------------------------------------*) (* Theorem 4.2 *) (* Chack: $$x = {t \over s} - 2s, \quad y = s t - {2 \over s}, \quad z = s^2 - 2t + {1 \over s^2}.$$ Eliminate $s$ and $t$ from these equations, then we have $\disc_4^+(x$, $y$, $z) = 0$ *) (* Factor[Disc4p[-(2s-t/s), s t-2/s, s^2 - 2t + 1/s^2]] *) Factor[Disc4p[-((-1 + 2 s^2 - s t)/s), (-2 + s^2 + s^3 t)/s, ((s^2 - 1)^2 - 2 s^3 t)/s^2]] (*---------------------------------------------------------------------------*) (* Theorem 4.3 *) (* Check: $\verline{E}$ is an irreducible sextic rational curve which can be represented as $$x = {1 \over 2}\left({1 \over s^3} - {r \over s} - 3s\right), \quad y = {1 \over 2}\left(s^3 - r s - {3 \over s}\right)$$ *) Factor[Disc4p[(1/2)(1/s^3 - r/s - 3s), (1/2)(s^3 - r s - 3/s), r]] (*---------------------------------------------------------------------------*) (* Theorem 4.7. *) (* Definition. Also use the above definitions. *) FrakE[a_,b_,c_,k_] := (k(a^2+b^2+c^2)-(a b+b c+c a))^2 (* Check ${\frak e}_k(a,b,c) := (k(a^2+b^2+c^2)-(ab+bc+ca))^2 = k^2 S_4 - 2k T_{3,1} + (2k^2+1)S_{2,2} - (2k-2)US_1 *) Factor[k^2 S4[a,b,c] - 2k T31[a,b,c] + (2k^2+1)S22[a,b,c] - (2k-2)US1[a,b,c] - FrakE[a,b,c,k]] (* Chack ${\frak e}_k(s,t,1) = 0$ *) Factor[FrakE[s,t,1,(s t+s+t)/(s^2+t^2+1)]] (* Check $k = - {3(p+q+1) \over p^2 + p q + q^2 + 3p + 3q + 9}$ *) Rel49[p_,q_,k_] := k (p^2 + p q + q^2 + 3p + 3q + 9) + 3(p+q+1) Factor[Rel49[FrakP[s,t], FrakQ[s,t], (s t+s+t)/(s^2+t^2+1)]] (*---------------------------------------------------------------------------*) (* Proof of 0.3 *) (* Definition. *) Disc4[p_,q_,r_,v_] := (3(r+1) - (p^2 + p q + q^2)) (2 p + 2 q + r + 5)^3 + v (p^4 + q^4 + 34 p^3 q + 34 p q^3 + 39 p^2 q^2 + 2 (p+q)(5p^2 + 7p q + 5q^2)r - (2 p^2 + p q + 2 q^2)r^2 + 86 p^3 + 86 q^3 - 12(v-16)(p^2 q + p q^2) - (v-84) (p^2+q^2) r + (v+18) p q r - 22 (p+q) r^2 + 8 r^3 - 57(v-2)(p^2+q^2) + (v^2-63v+51)p q - 2(13v+126)(p+q)r + 2(3v-106)r^2 + 2(7v^2 + 3v - 139)(p + q) + 8(19v - 70)r - (v^3 + 20v^2 - 162v + 388)) FrakR[s_,t_] := (FrakP[s,t]^2 + FrakP[s,t] FrakQ[s,t] + FrakQ[s,t]^2)/3 - 1 FrakV[s_,t_] := - (FrakP[s,t]^2 + FrakP[s,t] FrakQ[s,t] + FrakQ[s,t]^2)/3 - FrakP[s,t] - FrakQ[s,t] kw[s_,t_] := S11[s,t,1]/S2[s,t,1] pw[s_,t_,a2_,b2_] := (a2 FrakP[s,t] - 2 b2 kw[s,t])/(a2 + b2 kw[s,t]^2) qw[s_,t_,a2_,b2_] := (a2 FrakQ[s,t] - 2 b2 kw[s,t])/(a2 + b2 kw[s,t]^2) rw[s_,t_,a2_,b2_] := (a2 FrakR[s,t] + b2 (2 kw[s,t]^2+1))/(a2 + b2 kw[s,t]^2) (* vw[s_,t_,a2_,b2_] := (a2 FrakV[s,t] - b2 (2 kw[s,t] - 2))/(a2 + b2 kw[s,t]^2) *) vw2[s_,t_,a2_,b2_] := 3 b2 (kw[s,t] - 1)^2/(a2 + b2 kw[s,t]^2) (* Check $disc_4$ is a discriminant. *) Factor[Disc4[pw[s,t,1-u,u], qw[s,t,1-u,u], rw[s,t,1-u,u], vw2[s,t,1-u,u]]] (* Conjecture 4.11. *) ContourPlot3D[Disc4[p, q, r, 2000] == 0, {p, -200, 400}, {q, -200, 400}, {r, -1000, 2000}] ContourPlot3D[Disc4[p, q, r, 2] == 0, {p, -15, 20}, {q, -15, 20}, {r, -20, 20}] ContourPlot[Disc4[p, p, r, 2] == 0, {p, -10, 5}, {r, -5, 20}] ContourPlot3D[Disc4[p, p, r, v] == 0, {p, -10, 5}, {r, -2, 15}, {v, 0, 10}] ContourPlot3D[Disc4[p,q,r,200] == 0, {p,-30,30}, {q,-30,30}, {r,-10 200}] ContourPlot[Disc4[p,p,r,200] == 0, {p,-30,30}, {r,-10,200}] (*---------------------------------------------------------------------------*) (* Theorem 4.9. *) (* Definition. Also use the above definitions. *) varphif[x_,p_,q_,r_,v_] := 2 Sqrt[1+p+q+r+v] x^3 - (6+3p+3q+2r+v) x^2 + 2(1+p+q) Sqrt[1+p+q+r+v] x + 2+2r-v-(p^2+p q+q^2+p+q) f1[a_,b_,c_,p_,q_,r_,v_,x0_] := S4[a,b,c] + ((3p - 2 Sqrt[1+p+q+r+v] x0 + 2 x0^2)/(3-x0^2)) S31[a,b,c] + ((3q - 2 Sqrt[1+p+q+r+v] x0 + 2 x0^2)/(3-x0^2)) S13[a,b,c] + ((3r - (1+p+q+r+v) + 2 Sqrt[1+p+q+r+v] x0 - 3x0^2)/(3-x0^2)) S22[a,b,c] + ((3v - 2(1+p+q+r+v) + 2 Sqrt[1+p+q+r+v] x0)/(3-x0^2)) US1[a,b,c] (* Check $3f = (3-x_0^2) f_1 + (x_0-\sqrt{\alpha_f}\,)^2 e_1 \geq (3-x_0^2) f_1$ *) Factor[ 3(S4[a,b,c] + p S31[a,b,c] + q S13[a,b,c] + r S22[a,b,c] + v US1[a,b,c]) - (3-x0^2) f1[a,b,c,p,q,r,v,x0] - (x0 - Sqrt[1+p+q+r+v])^2 FrakE[a,b,c, x0/(x0-Sqrt[1+p+q+r+v])]] Factor[f1[a,a,a,p,q,r,v,x0]] (* Check ${p_1^2 + p_1 q_1 + q_1^2 \over 3} - 1 - r_1 = - {3 \varphi_f(x_0) \over (3-x_0^2)^2}$ *) Factor[(((3p - 2 Sqrt[1+p+q+r+v] x0 + 2 x0^2)/(3-x0^2))^2 + ((3p - 2 Sqrt[1+p+q+r+v] x0 + 2 x0^2)/(3-x0^2))((3q - 2 Sqrt[1+p+q+r+v] x0 + 2 x0^2)/(3-x0^2)) + ((3q - 2 Sqrt[1+p+q+r+v] x0 + 2 x0^2)/(3-x0^2))^2)/3 - 1 - (3r - (1+p+q+r+v) + 2 Sqrt[1+p+q+r+v] x0 - 3x0^2)/(3-x0^2) + 3 varphif[x0,p,q,r,v]/(3-x0^2)^2] (* Check $f = {3-x_0^2 \over 3} {\frak g}_{p_1,q_1} + {\varphi_f(x_0) \over 3-x_0^2} {\frak g}_{\infty} + {(x_0-\sqrt{\alpha_f}\,)^2 \over 3} e_1$ *) Factor[(S4[a,b,c] + p S31[a,b,c] + q S13[a,b,c] + r S22[a,b,c] + v US1[a,b,c]) - ((3-x0^2)/3) FrakG[a,b,c,(3p - 2 Sqrt[1+p+q+r+v] x0 + 2 x0^2)/(3-x0^2), (3q - 2 Sqrt[1+p+q+r+v] x0 + 2 x0^2)/(3-x0^2)] - (varphif[x0,p,q,r,v]/(3-x0^2)) (S22[a,b,c] - US1[a,b,c]) - ((x0 - Sqrt[1+p+q+r+v])^2/3) FrakE[a,b,c, x0/(x0-Sqrt[1+p+q+r+v])]] (* Check \varphi_f(\sqrt{3}) = -4 \left(\alpha - {\sqrt{3} \over 4}(p+q+4)\right)^2 - {(p-q)^2 \over 4} *) Factor[varphif[Sqrt[3],p,q,r,v] + 4(Sqrt[1+p+q+r+v] - (Sqrt[3]/4)(p+q+4))^2 + (p-q)^2/4] Factor[varphif[-Sqrt[3],p,q,r,v] + 4(Sqrt[1+p+q+r+v] + (Sqrt[3]/4)(p+q+4))^2 + (p-q)^2/4] (* Check $f = \alpha_2 {\frak g}_{p_2,q_2} + \beta_2 {\frak e}_k$. *) Factor[a2 FrakG[a,b,c,p2,q2] + b2 FrakE[a,b,c,k] - ((a2 + b2 k^2)S4[a,b,c] + (a2 p2 - 2 b2 k) S31[a,b,c] + (a2 q2 - 2 b2 k) S13[a,b,c] + (a2 ((p2^2+p2 q2 +q2^2)/3 - 1) + b2 (2k^2+1)) S22[a,b,c] - (a2 ((p2^2+p2 q2 +q2^2)/3 +p2 + q2) + b2 (2k-2)) US1[a,b,c])] (* Check $\left|{\sqrt{\alpha_f} k \over k-1}\right| = \left|-{\sqrt{3 \beta_2} k \over \sqrt{\alpha_2 + \beta_2 k^2}}\right| The following only calculates $\sqrt{\alpha_f} k \over k-1}$ *) Factor[Sqrt[1 + (a2 p2 - 2 b2 k)/(a2 + b2 k^2) + (a2 q2 - 2 b2 k)/(a2 + b2 k^2) + (a2 ((p2^2+p2 q2 +q2^2)/3 - 1) + b2 (2k^2+1))/(a2 + b2 k^2) - (a2 ((p2^2+p2 q2 +q2^2)/3 +p2 + q2) + b2 (2k-2))/(a2 + b2 k^2)] k/(k-1)] (* Check $\varphi_f\left({\sqrt{\alpha_f} k \over k-1}\right) = 0$ *) varphifw[a2_,b2_,p2_,q2_,k_] := varphif[Sqrt[1 + (a2 p2 - 2 b2 k)/(a2 + b2 k^2) + (a2 q2 - 2 b2 k)/(a2 + b2 k^2) + (a2 ((p2^2+p2 q2 +q2^2)/3 - 1) + b2 (2k^2+1))/(a2 + b2 k^2) - (a2 ((p2^2+p2 q2 +q2^2)/3 +p2 + q2) + b2 (2k-2))/(a2 + b2 k^2)] k/(k-1), (a2 p2 - 2 b2 k)/(a2 + b2 k^2), (a2 q2 - 2 b2 k)/(a2 + b2 k^2), (a2 ((p2^2+p2 q2 +q2^2)/3 - 1) + b2 (2k^2+1))/(a2 + b2 k^2), -(a2 ((p2^2+p2 q2 +q2^2)/3 +p2 + q2) + b2 (2k-2))/(a2 + b2 k^2)] Factor[varphifw[a2,b2,p2,q2,k]] (* Check that if $k = {S_{1,1}(s,t,1) \over S_2(s,t,1)}$, then $x = {\sqrt{\alpha_f} k \over k-1}$ is a multiple root of $\varphi_f(x)$. We chak $\varphi_f(x) = (x - {\sqrt{\alpha_f} k \over k-1})^2 varphif2[x,s,t,a2,b2]$ *) FrakR[s_,t_] := (FrakP[s,t]^2 + FrakP[s,t] FrakQ[s,t] + FrakQ[s,t]^2)/3 - 1 FrakV[s_,t_] := - (FrakP[s,t]^2 + FrakP[s,t] FrakQ[s,t] + FrakQ[s,t]^2)/3 - FrakP[s,t] - FrakQ[s,t] kw[s_,t_] := S11[s,t,1]/S2[s,t,1] pw[s_,t_,a2_,b2_] := (a2 FrakP[s,t] - 2 b2 kw[s,t])/(a2 + b2 kw[s,t]^2) qw[s_,t_,a2_,b2_] := (a2 FrakQ[s,t] - 2 b2 kw[s,t])/(a2 + b2 kw[s,t]^2) rw[s_,t_,a2_,b2_] := (a2 FrakR[s,t] + b2 (2 kw[s,t]^2+1))/(a2 + b2 kw[s,t]^2) vw[s_,t_,a2_,b2_] := (a2 FrakV[s,t] - b2 (2 kw[s,t] - 2))/(a2 + b2 kw[s,t]^2) alphaw[s_,t_,a2_,b2_] := Sqrt[1 + pw[s,t,a2,b2] + qw[s,t,a2,b2] + rw[s,t,a2,b2] + vw[s,t,a2,b2]] Factor[varphif[alphaw[s,t,a2,b2] kw[s,t]/(kw[s,t]-1), pw[s,t,a2,b2], qw[s,t,a2,b2], rw[s,t,a2,b2], vw[s,t,a2,b2]]] varphifc[s_,t_,a2_,b2_] := 2 Sqrt[3 b2 (1 - s + s^2 - t - s t + t^2)^2 /( a2 + 2 a2 s^2 + b2 s^2 + a2 s^4 + 2 b2 s t + 2 b2 s^2 t + 2 a2 t^2 + b2 t^2 + 2 b2 s t^2 + 2 a2 s^2 t^2 + b2 s^2 t^2 + a2 t^4)] varphif2[x_,s_,t_,a2_,b2_]:= varphifc[s,t,a2,b2] x - ((1 - s + s^2 - t - s t + t^2) (a2 - a2 s + 3 a2 s^2 + 4 b2 s^2 - 2 a2 s^3 + 2 b2 s^3 + 3 a2 s^4 + 4 b2 s^4 - a2 s^5 + a2 s^6 - a2 t - a2 s t - 4 b2 s t - 2 a2 s^2 t - 4 b2 s^2 t - 2 a2 s^3 t - 4 b2 s^3 t - a2 s^4 t - 4 b2 s^4 t - a2 s^5 t + 3 a2 t^2 + 4 b2 t^2 - 2 a2 s t^2 - 4 b2 s t^2 + 6 a2 s^2 t^2 + 6 b2 s^2 t^2 - 2 a2 s^3 t^2 - 4 b2 s^3 t^2 + 3 a2 s^4 t^2 + 4 b2 s^4 t^2 - 2 a2 t^3 + 2 b2 t^3 - 2 a2 s t^3 - 4 b2 s t^3 - 2 a2 s^2 t^3 - 4 b2 s^2 t^3 - 2 a2 s^3 t^3 + 2 b2 s^3 t^3 + 3 a2 t^4 + 4 b2 t^4 - a2 s t^4 - 4 b2 s t^4 + 3 a2 s^2 t^4 + 4 b2 s^2 t^4 - a2 t^5 - a2 s t^5 + a2 t^6))/((s^2 - s t - s^2 t + t^2 - s t^2 + s^2 t^2) (a2 + 2 a2 s^2 + b2 s^2 + a2 s^4 + 2 b2 s t + 2 b2 s^2 t + 2 a2 t^2 + b2 t^2 + 2 b2 s t^2 + 2 a2 s^2 t^2 + b2 s^2 t^2 + a2 t^4)) Factor[varphif[x, pw[s,t,a2,b2], qw[s,t,a2,b2], rw[s,t,a2,b2], vw[s,t,a2,b2]] - (x - alphaw[s,t,a2,b2] kw[s,t]/(kw[s,t]-1))^2 varphif2[x,s,t,a2,b2]] (*--------------------------------------------------------------------------*) (* Theorem 4.10. *) (* Check $$(s^2 + 1)^2 e_s + 3 s {\frak h}_s = s^2 g_s + 3(s^2-s+1)^2 US_1$$ *) Gs[a_,b_,c_,s_] := FrakG[a,b,c,FrakP[0,s],FrakQ[0,s]] Es[a_,b_,c_,s_] := FrakE[a,b,c,s/(1+s^2)] Factor[s^2 Gs[a,b,c,s] - FrakK[a,b,c,s,1]] Factor[(s^2 + 1)^2 Es[a,b,c,s] + 3 s FrakH[a,b,c,s] - s^2 Gs[a,b,c,s] - 3(s^2-s+1)^2 US1[a,b,c]] (*---------------------------------------------------------------------------*) (* Deleted Theorem. *) (* Part (II-a) *) (* Check $2 s^4 + p s^3 - q s - 2 = 0$. *) Eliminate[{1 == alpha + beta s^2/(s^2+1)^2, p == alpha (1-2s^2)/s - 2 beta s / (s^2+1), q == alpha (s^2 - 2)/s - 2 beta s / (s^2+1)}, {alpha, beta}] (* Check $\alpha = {2s^2 + p s + 2 \over 3}$, $\beta = - {(s^2+1)^2(2s^2 + ps - 1) \over 3 s^2}$. *) Solve[{1 == alpha + beta s^2/(s^2+1)^2, p == alpha (1-2s^2)/s - 2 beta s / (s^2+1)}, {alpha, beta}] (* Check $f = {2s^2+ p s +2 \over 3} g_s - {(s^2+1)^2(2s^2 + ps - 1) \over 3 s^2} e_s + \gamma_3 (S_{2,2}-US_1) + (\gamma_3+\delta_3) US_1$ *) (* \gamma_3 := r + 3 s^2 + 2 p s - 1/s^2 \delta_3 := v + (2s+p)(s-1)(s^2+1) + p/s *) Factor[(S4[a,b,c] + p S31[a,b,c] + (2 s^3 + p s^2 - 2/s) S13[a,b,c] + r S22[a,b,c] + v US1[a,b,c]) - ( ((2s^2+ p s +2)/3) Gs[a,b,c,s] - ((s^2+1)^2(2s^2 + p s - 1)/(3 s^2)) Es[a,b,c,s] + (r + 3 s^2 + 2 p s - (1/s^2))(S22[a,b,c] - US1[a,b,c]) + ((r + 3 s^2 + 2 p s - (1/s^2)) + (v + (2s+p)(s-1)(s^2+1) + p/s)) US1[a,b,c])] (* Part (II-b) *) (* Check: $p = - 2s + {t \over s}$, $q = s t - {2 \over s}$. Eliminate $t$ from the above two equations, we have $2s^4 + p s^3 - q s - 2 = 0$. *) Eliminate[{p == - 2s + t/s, q == s t - 2/s}, t] (* Check: $f - {1 \over s^2} {\frak k}_{s,t} = \gamma_4 S_{2,2} + \delta_4 US_1$. *) (* \gamma_4 := r + 3 s^2+ 2 p s - 1/s^2 \delta_4 := v + (s-1)^2p + (2 s^5 - 3 s^4 + s^2 - 2 s + 1)/s^2 *) Factor[(S4[a,b,c] + p S31[a,b,c] + (2 s^3 + p s^2 - 2/s) S13[a,b,c] + r S22[a,b,c] + v US1[a,b,c]) - 1/s^2 FrakK[a,b,c,s, (2s^2 + p s)] - (r + 3 s^2 + 2 p s - 1/s^2) S22[a,b,c] - (v + (s-1)^2p + (2 s^5 - 3 s^4 + s^2 - 2 s + 1)/s^2) US1[a,b,c]] (*---------------------------------------------------------------------------*) (* Lemma 4.14. *) g418[x_,t_] := 108 Sqrt[3] t^3 + 36(10-3x)t^2 - Sqrt[3](x+2)^2(4x+47)t + 6(x+2)^4 h418[x_,v_] := (4v + x^2 - 44x + 52)^2 + 128 (x-4)^3 Dg418[x_,t_] := 324 Sqrt[3] t^2 + 72(10-3x)t - Sqrt[3](x+2)^2(4x+47) g1[x_] := 12 x^3 + 225 x^2 + 372 x + 964 g2[x_] := 378 x^4 + 2331 x^3 + 13986 x^2 + 21636 x + 32696 t1[x_] := (Sqrt[g1[x]] - 2 (10 - 3 x))/(18 Sqrt[3]) (* Check: g(x,t_1(x)) = {1 \over 81}\big(g_2(x) - g_1(x)^{3/2} \big) *) Factor[g418[x,t1[x]] - (1/81)(g2[x]-g1[x] Sqrt[g1[x]])] (* Check $g_2(x)^2 - g_1(x)^3 = 108 (12 - x)^3 (x + 2)^4 (16x^2 + 25 x + 58)$ *) Factor[g2[x]^2 - g1[x]^3 - (108 (12 - x)^3 (x + 2)^4 (16x^2 + 25 x + 58))] Factor[g418[x,Sqrt[3]]] (* Check h(x,y) = (x-4)^2\big((x+24)^2 + 512\big) + 8 (v-27)\big(2(v-27) + (x-4)(x-40) \big) *) Factor[h418[x,v] - ((x-4)^2((x+24)^2+512) + 8(v-27)(2(v-27)+(x-4)(x-40)))] (* Theorem 4.19. *) Factor[Disc4[x, x, (x^2 - (t^2 + 2 Sqrt[3] t + 1)), t^2] - (-Sqrt[3] t (x + (2 Sqrt[3]/3) t + 2)^2 g418[x, t])] Factor[Disc4[x, x, (x^2/4) + 2, v] - (-((3(x+2)^2 - 4 v)^2 h418[x,v]/256))] (*---------------------------------------------------------------------------*) (* Proposition 5.2. *) 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] f52[a_,b_,c_,p_,q_,r_] := S5[a,b,c] + p T41[a,b,c] + q T32[a,b,c] + r US2[a,b,c] - (1+2p+2q+r) US11[a,b,c] (* Calculate f(x,1,0) *) Factor[f52[x,1,0,p,q,r]] Eliminate[{x==s1[s,t,1]/s0[s,t,1], y==s2[s,t,1]/s0[s,t,1], z==s3[s,t,1]/s0[s,t,1]}, {s,t}] Eliminate[{x0==s0[a,b,c], x1==s1[a,b,c], x2==s2[a,b,c], x3==s3[a,b,c]}, {a,b,c}] (*---------------------------------------------------------------------------*) (* Proposition 5.3. *) (* Calculate f(x,1,1) *) g53[x_,p_,q_,r_] := x^3 + 2(p+1)x^2+(4p+2q+r+3)x + 2(p+q+1) Dg53[x_,p_,q_,r_] := 3x^2 + 4(p+1)x + (4p+2q+r+3) D53[p_,q_,r_] := 4(p+1)(p-2)(2p-1) - 9(q(2p-1)+r(p+1)) - Sqrt[(2p-1)^2 - 3(2q+r+2)]^3 Factor[f52[x,1,1,p,q,r]] Factor[f52[x,1,1,p,q,r] - (x-1)^2 g53[x,p,q,r]] (* Check $g'(-{2(p+1) \over 3}\right) = {3(2q+r+2) - (2p-1)^2 \over 3}$ *) Factor[Dg53[-2(p+1)/3,p,q,r] - (3(2q+r+2) - (2p-1)^2)/3] (* Chaek $x_0 = {\sqrt{(2p-1)^2 - 3(2q+r+2)} - 2(p+1) \over 3}$ *) Solve[Dg53[x,p,q,r]==0, x] Factor[Dg53[(Sqrt[(2p-1)^2 - 3(2q+r+2)] - 2(p+1))/3, p,q,r]] (* Check $27 g(x_0) = 2d(p,q,r)$ *) Factor[27 g53[(Sqrt[(2p-1)^2 - 3(2q+r+2)] - 2(p+1))/3, p,q,r] - 2 D53[p,q,r]] (*---------------------------------------------------------------------------*) (* Proof of Theorem 0.4. *) (* Check: The following parameterlization define Disk5[p,q,r] *) (* q = {1 \over 27 (2t + 1)^3} \cdot \bigg(-3^6 t^3 + (p+1)\Big(4(8p^2-65p+116)t^3 + 6(8p^2-38p-19)t^2 + 3(8p^2-11p-73)t + (4p^2+8p-23)\Big)\bigg), r = {1 \over 27 (2t + 1)^3} \cdot \Big(-8(2p-1)^3 t^3 - 3(2p-1)^2(8p+23) t^2 - 6(2p-1)(p+4)(4p+7)t - (p+4)^2(8p + 5) \Big) *) D5[p_,q_,r_] := (4(p+1)(p-2)(2p-1) - 9q(2p-1) - 9r(p+1))^2 - ((2p-1)^2 - 3(2q+r+2))^3 q54[t_,p_] := (1/(27 (2t + 1)^3)) (-3^6 t^3 + (p+1)(4(8p^2-65p+116)t^3 + 6(8p^2-38p-19)t^2 + 3(8p^2-11p-73)t + (4p^2+8p-23))) r54[t_,p_] := (1/(27 (2t + 1)^3)) (-8(2p-1)^3 t^3 - 3(2p-1)^2(8p+23) t^2 - 6(2p-1)(p+4)(4p+7)t - (p+4)^2(8p + 5)) Factor[D5[p, q54[t,p], r54[t,p]]] (* Check: $P_p = ({4p^3 + 12 p^2 - 15 p - 23 \over 27}, -{(2p-1)^3 \over 27}) lies on D5[p,q,r] *) Sing54q[p_] := (4 p^3 + 12 p^2 - 15 p - 23)/27 Sing54r[p_] := -(2 p - 1)^3/27 Factor[D5[p,Sing54q[p],Sing54r[p]]] ParametricPlot[{q54[1/t, 1], r54[1/t, 1]}, {t, -1, 1}] ParametricPlot[{q54[1/t, 0], r54[1/t, 0]}, {t, -1, 5}] ParametricPlot[{q54[t, -1], r54[t, -1]}, {t, -0.2, 2}] ParametricPlot[{q54[t, -2], r54[t, -2]}, {t, -0.4, 3}] (*--------------------------------------------------------------------------*) (* Proposition 5.5. *) (* Check Q_{s,t}(s,1,1) = 0 *) Q55[a_,b_,c_,s_,t_] := (S5[a,b,c] + (s^3 + s^2 - 1) T32[a,b,c] - (2 s^3 + 5 s^2 + 4 s + 1) US2[a,b,c] + (3 s^2 + 4 s + 2) US11[a,b,c]) + t (T41[a,b,c] + (s^2 - 1) T32[a,b,c] - 2 (s + 1)^2 US2[a,b,c] + (4 s + 2) US11[a,b,c]) Factor[Q55[s,1,1,s,t]] (*--------------------------------------------------------------------------*) (* Theorem 5.6. *) F591[a_,b_,c_,s_] := 3s^4 S5[a,b,c] - (4s^5-1)S41[a,b,c] + (s^8-4s^3)S14[a,b,c]-(s^8-4s^5+3s^4-4s^3+1)US2[a,b,c] F592[a_,b_,c_,s_] := 2 S41[a,b,c] + s^3 S14[a,b,c] - 3s S32[a,b,c]-(s^3-3s+2)US2[a,b,c] F593[a_,b_,c_,s_] := S41[a,b,c] + 2s^3 S14[a,b,c] - 3s^2 S23[a,b,c]-(2s^3-3s^2+1)US2[a,b,c] F594[a_,b_,c_,s_] := US2[a,b,c] - US11[a,b,c] (* Check: $F_{i,s}(s,1,0) = 0$ *) Factor[F591[s,1,0,s]] Factor[F592[s,1,0,s]] Factor[F593[s,1,0,s]] Factor[F594[s,1,0,s]] (* Check $F_{3,s}(a,b,c) = s^3 F_{2,s}(b,a,c,1/s)$ *) Factor[F593[a,b,c,s] - s^3 F592[b,a,c,1/s]] (*-------------------------------------------------------------------------------*) (* Theorem 5.9. *) (* Check: $\disc_5^+$ is adiscriminant. *) Disc5p[x_,y_,z_,w_] := - 27 x^4 y^4 - 4 x^3 y^2 w^3 - 4 x^2 y^3 z^3 + 18 x^3 y^3 z w + x^2 y^2 z^2 w^2 + 144 x^4 y^2 w + 144 x^2 y^4 z - 6 x^3 y^2 z^2 - 6 x^2 y^3 w^2 + 16 x^3 w^4 + 16 y^3 z^4 - 80 x^3 y z w^2 - 80 x y^3 z^2 w + 18 x^2 y z^3 w + 18 x y^2 z w^3 - 4 x^2 z^2 w^3 - 4 y^2 z^3 w^2 - 36 x^3 y^3 - 192 x^4 y z - 192 x y^4 w - 128 x^4 w^2 - 128 y^4 z^2 + 24 x^2 y w^3 + 24 x y^2 z^3 - 27 x^2 z^4 - 27 y^2 w^4 - 746 x^2 y^2 z w + 144 x^3 z^2 w + 144 y^3 z w^2 - 72 x z w^4 - 72 y z^4 w + 356 x y z^2 w^2 + 16 z^3 w^3 + 256 x^5 + 256 y^5 + 160 x^3 y w + 160 x y^3 z + 1020 x^2 y z^2 + 1020 x y^2 w^2 + 560 x^2 z w^2 + 560 y^2 z^2 w - 630 x z^3 w - 630 y z w^3 + 108 z^5 + 108 w^5 - 50 x^2 y^2 - 1600 x^3 z - 1600 y^3 w - 900 x w^3 - 900 y z^3 - 2050 x y z w + 825 z^2 w^2 + 2000 x^2 w + 2000 y^2 z + 2250 x z^2 + 2250 y w^2 - 2500 x y - 3750 z w + 3125 Factor[Disc5p[-(4s^5 - 1)/(3s^4) + 2 alpha2 + alpha3, (s^8 - 4s^3)/(3s^4) + s^3 alpha2 + 2s^3 alpha3, -3s alpha2, -3s^2 alpha3]] Factor[Disc5p[p,p,q,q]] (*---------------------------------------------------------------------------*) (* Theorem 6.1. *) 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] (* !! How to use this tool !! *) (* We use this tool to prove that given sextic homogeneous symmetric inequality. You must define Ftest[a,b,c] which you want to prove. The follwoing Ftest[a,b,c] is a sample (Corollary 5.2(1)). *) Ftest[a_, b_, c_] := 6 S6[a,b,c] - 8 T51[a,b,c] + 5 T42[a,b,c] + 0 S33[a,b,c] + 0 UT21[a,b,c] + 0 U2[a,b,c] Solve[Eliminate[{Ftest[a, b, c] == f, p == a + b + c, q == a b + b c + c a, r == a b c}, {a, b, c}],f] (* Using the above result. You must represent Ftest[a,b,c] as the form Ftest[a,b,c] = g0[p,q] r^2 + g1[p,q] r + g2[p, q] =: Htest[p_,q_,r_] where $p = a+b+c$, $q = a b+b c+c a$, $r = a b c$ *) Htest[p_,q_,r_] := (* You must write here *) (* In the case the above Ftest[a,b,c], Htest[p,q,r] is as the following. *) Htest[p_,q_,r_] := 6 p^6 - 44 p^4 q + 91 p^2 q^2 - 38 q^3 + 34 p^3 r - 108 p q r + 27 r^2 (* After you define Htest[p,q,r], do the following. *) (* Dtest[p,q], H1test[s], H2test[t] are $D(p,q)$, $h_1(s)$, $h_2(t)$ in Theorem 6.1 *) G2[p_,q_]:=Htest[p,q,0] A:=(Htest[p,q,1]+Htest[p,q,-1])/2 - Htest[p,q,0] G1[p_,q_]:=Htest[p,q,1]-G2[p,q]-A DD[p_,q_]:=G1[p,q]^2 - 4 A G2[p,q] H1[s_] := 2 A s + G1[s + 2, 2 s + 1] H2[t_] := 2 A t^2 + G1[2 t + 1, t^2 + 2 t] Expand[A] Expand[G1[p,q]] Expand[G2[p,q]] Expand[DD[p,q]] Factor[DD[p,q]]//N Factor[DD[s+2,2s+1]]//N Solve[DD[s+2,2s+1]==0]//N Factor[DD[2t+1,2t+t^2]]//N Solve[DD[2t+1,2t+t^2]==0]//N Factor[G[x, 1, 0]]//N Solve[G[x, 1, 0] == 0] // N Factor[G[x, 1, 1]]//N Solve[G[x, 1, 1] == 0] // N Factor[H1[s]]//N Factor[H2[(1+2s)/(4-s)]]//N Factor[H2[t]]//N (*---------------------------------------------------------------------------*) (* Check the proof of Theorem 6.1 *) alpha1[p_,q_] := (p - 2 Sqrt[p^2 - 3q])/3 alpha2[p_,q_] := (p + 2 Sqrt[p^2 - 3q])/3 beta1[p_,q_] := (p - Sqrt[p^2 - 3q])/3 beta2[p_,q_] := (p + Sqrt[p^2 - 3q])/3 r1[p_,q_] := alpha1[p,q] beta2[p,q]^2 r2[p_,q_] := beta1[p,q]^2 alpha2[p,q] (*---------------------------------------------------------------------------*) (* Lemma 6.6. *) 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] G164[a_,b_,c_,s_] := s^4 S6[a,b,c] - (2s^6-1) S42[a,b,c] + (s^8-2s^2) S24[a,b,c] - 3 (s^8-2s^6+s^4-2s^2+1) U2[a,b,c] G264[a_,b_,c_,s_] := 2 S51[a,b,c] - 3 s S42[a,b,c] + s^3 S24[a,b,c] - 3 (s-1)^2(s+2) U2[a,b,c] G364[a_,b_,c_,s_] := 2s^3 S15[a,b,c] + S42[a,b,c] - 3s^2 S24[a,b,c] - 3(s-1)^2(2s+1) U2[a,b,c] G464[a_,b_,c_,s_] := S42[a,b,c] + s^2 S24[a,b,c] - 2 s S33[a,b,c] - 3 (s-1)^2 U2[a,b,c] G564[a_,b_,c_,s_] := US3[a,b,c] - US21[a,b,c] G664[a_,b_,c_,s_] := US3[a,b,c] - US12[a,b,c] G764[a_,b_,c_,s_] := US3[a,b,c] - 3 U2[a,b,c] (* Check: $G_{i,s}(s,1,0) = 0$. *) Factor[G164[s,1,0,s]] Factor[G264[s,1,0,s]] Factor[G364[s,1,0,s]] Factor[G464[s,1,0,s]] Factor[G564[s,1,0,s]] Factor[G664[s,1,0,s]] Factor[G764[s,1,0,s]] (* Check: $G_{1,s}(a,b,c) = {\frak f}_{s^2}(a^2, b^2, c^2)$ *) Factor[G164[a,b,c,s] - FrakF[a^2,b^2,c^2,s^2]] (* Check: $G_{3,s} \geq 0$ follows from $G_{3,s}(a,b,c) = s^3 G_{2,1/s}(b,a,c)$. *) Factor[G364[a, b, c, s] - s^3 G264[b, a, c, 1/s]] (* Check G_{4,s} = (S_{2,4}- 3U^2) \left\{ \left(s - {S_{3,3} - 3U^2 \over S_{2,4} - 3U^2} \right)^2 + {U^2(S_6 + 6S_{3,3} + 3U^2 - 3T_{4,2} - 2US_3) \over (S_{2,4}-3U^2)^2}\right\}. *) Factor[G464[a,b,c,s] - (S24[a,b,c]- 3U2[a,b,c]) ((s - (S33[a,b,c] - 3U2[a,b,c])/(S24[a,b,c] - 3U2[a,b,c]))^2 + U2[a,b,c](S6[a,b,c] + 6S33[a,b,c] + 3U2[a,b,c] - 3T42[a,b,c] - 2US3[a,b,c])/( S24[a,b,c]-3U2[a,b,c])^2)] (*--------------------------------------------------------------------------*) (* Theorem 6.7. *) Disc6p[x_,y_,z_,w_,u_] := 256 x^5 y^5 - 27 x^4 y^2 w^4 - 27 x^2 y^4 z^4 - 192 x^4 y^4 z w - 6 x^3 y^3 z^2 w^2 - 4 x^2 y^2 z^3 w^3 + 144 x^4 y^3 w^2 u + 144 x^3 y^4 z^2 u + 18 x^3 y^2 z w^3 u + 18 x^2 y^3 z^3 w u - 128 x^4 y^4 u^2 - 80 x^3 y^3 z w u^2 + x^2 y^2 z^2 w^2 u^2 - 4 x^3 y^2 w^2 u^3 - 4 x^2 y^3 z^2 u^3 + 16 x^3 y^3 u^4 - 1600 x^3 y^5 z - 1600 x^5 y^3 w - 36 x^3 y^3 z^3 - 36 x^3 y^3 w^3 + 108 x^4 w^5 + 108 y^4 z^5 + 1020 x^4 y^2 z w^2 + 1020 x^2 y^4 z^2 w + 24 x^3 y z^2 w^3 + 24 x y^3 z^3 w^2 + 16 x^2 z^3 w^4 + 16 y^2 z^4 w^3 + 144 x^2 y^2 z^4 w + 160 x^4 y^3 z u + 160 x^3 y^4 w u - 630 x^4 y w^3 u - 630 x y^4 z^3 u - 746 x^3 y^2 z^2 w u - 746 x^2 y^3 z w^2 u - 72 x^3 z w^4 u - 72 y^3 z^4 w u - 80 x^2 y z^3 w^2 u - 80 x y^2 z^2 w^3 u + 560 x^4 y^2 w u^2 + 560 x^2 y^4 z u^2 + 356 x^3 y z w^2 u^2 + 356 x y^3 z^2 w u^2 - 6 x^2 y^2 z^3 u^2 - 6 x^2 y^2 w^3 u^2 - 4 x^2 z^2 w^3 u^2 - 4 y^2 z^3 w^2 u^2 + 24 x^3 y^2 z u^3 + 24 x^2 y^3 w u^3 + 16 x^3 w^3 u^3 + 16 y^3 z^3 u^3 + 18 x^2 y z^2 w u^3 + 18 x y^2 z w^2 u^3 - 72 x^3 y w u^4 - 72 x y^3 z u^4 + 320 x^4 y^4 - 50 x^4 y^2 z^2 - 50 x^2 y^4 w^2 + 2250 x y^5 z^2 + 2250 x^5 y w^2 + 144 x^3 y w^4 + 144 x y^3 z^4 + 9768 x^3 y^3 z w + 160 x^3 y z^3 w + 160 x y^3 z w^3 - 900 x^4 z w^3 - 900 y^4 z^3 w - 576 x^2 z w^5 - 576 y^2 z^5 w - 5428 x^2 y^2 z^2 w^2 - 128 x^2 z^4 w^2 - 128 y^2 z^2 w^4 - 96 x y z^3 w^3 + 144 x^2 y^2 z w^4 - 64 z^4 w^4 + 2000 x^5 y^2 u + 2000 x^2 y^5 u - 2050 x^4 y z w u - 2050 x y^4 z w u - 682 x^3 y^2 w^2 u - 682 x^2 y^3 z^2 u - 192 x^2 y z^4 u - 192 x y^2 w^4 u + 3272 x^2 y z w^3 u + 3272 x y^2 z^3 w u + 320 x z^2 w^4 u + 320 y z^4 w^2 u - 208 x^3 y^3 u^2 + 825 x^4 w^2 u^2 + 825 y^4 z^2 u^2 + 1020 x^3 y z^2 u^2 + 1020 x y^3 w^2 u^2 + 24 x^2 w^4 u^2 + 24 y^2 z^4 u^2 + 144 x^2 z^3 w u^2 + 144 y^2 z w^3 u^2 - 1584 x y z^2 w^2 u^2 + 16 z^3 w^3 u^2 - 900 x^4 y u^3 - 900 x y^4 u^3 - 630 x^3 z w u^3 - 630 y^3 z w u^3 - 108 x^2 y w^2 u^3 - 108 x y^2 z^2 u^3 - 72 x z w^3 u^3 - 72 y z^3 w u^3 - 27 x^2 z^2 u^4 - 27 y^2 w^2 u^4 + 324 x y z w u^4 + 108 x^3 u^5 + 108 y^3 u^5 - 2500 x^5 y z - 2500 x y^5 w - 1700 x^2 y^4 z - 1700 x^4 y^2 w + 248 x^2 y^2 z^3 + 248 x^2 y^2 w^3 + 256 x^2 z^5 + 256 y^2 w^5 + 2000 x^4 z^2 w + 2000 y^4 z w^2 - 13040 x^3 y z w^2 - 13040 x y^3 z^2 w+ 4816 x^2 z^2 w^3 + 4816 y^2 z^3 w^2 + 512 z^5 w^2 + 512 z^2 w^5 - 640 x y z w^4 - 3750 x^5 w u - 3750 y^5 z u - 12330 x^3 y^2 z u - 12330 x^2 y^3 w u - 1600 x^3 z^3 u - 1600 y^3 w^3 u - 120 x^3 w^3 u - 120 y^3 z^3 u + 560 x^3 z^2 w^2 u + 560 y^3 z^2 w^2 u + 10152 x^2 y z^2 w u + 10152 x y^2 z w^2 u + 768 x w^5 u + 768 y z^5 u - 2496 x z^3 w^2 u - 2496 y z^2 w^3 u + 2250 x^4 z u^2 + 2250 y^4 w u^2 + 1980 x^3 y w u^2 + 1980 x y^3 z u^2 - 4536 x^2 z w^2 u^2 - 4536 y^2 z^2 w u^2 - 4464 x y z^3 u^2 - 4464 x y w^3 u^2 - 576 z^4 w u^2 - 576 z w^4 u^2 + 3942 x^2 y z u^3 + 3942 x y^2 w u^3 + 2808 x z^2 w u^3 + 2808 y z w^2 u^3 + 162 x^2 w u^4 + 162 y^2 z u^4 + 108 z^3 u^4 + 108 w^3 u^4 - 486 x z u^5 - 486 y w u^5 + 3125 x^6 + 410 x^3 y^3 + 3125 y^6 + 15600 x^3 y z^2 + 15600 x y^3 w^2 + 1500 y^4 z^2 + 1500 x^4 w^2 - 192 x^2 w^4 - 192 y^2 z^4 - 10560 x^2 z^3 w - 10560 y^2 z w^3 + 8748 x^2 y^2 z w - 640 x y z^4 w + 15264 x y z^2 w^2 - 1024 z^6 - 4352 z^3 w^3 - 1024 w^6 + 2250 x^4 y u + 2250 x y^4 u + 19800 x^3 z w u + 19800 y^3 z w u + 16632 x^2 y w^2 u + 16632 x y^2 z^2 u + 6912 x z^4 u + 6912 y w^4 u - 5760 x z w^3 u - 5760 y z^3 w u + 15417 x^2 y^2 u^2 - 2412 x^2 y^2 z w u^2 - 9720 x^2 z^2 u^2 - 9720 y^2 w^2 u^2 - 22896 x y z w u^2 + 8208 z^2 w^2 u^2 - 1350 x^3 u^3 - 1350 y^3 u^3 + 5832 x w^2 u^3 + 5832 y z^2 u^3 - 6318 x y u^4 - 4860 z w u^4 + 729 u^6 - 22500 x^4 z - 22500 y^4 w - 1800 x y^3 z - 1800 x^3 y w - 21888 x y z^3 - 21888 x y w^3 - 6480 x^2 z w^2 - 6480 y^2 z^2 w + 9216 z^4 w + 9216 z w^4 - 31320 x^2 y z u - 31320 x y^2 w u - 3456 x z^2 w u - 3456 y z w^2 u - 27540 x^2 w u^2 - 27540 y^2 z u^2 - 8640 z^3 u^2 - 8640 w^3 u^2 + 21384 x z u^3 + 21384 y w u^3 + 540 x^2 y^2 + 43200 x^2 z^2 + 43200 y^2 w^2 + 31968 x y z w - 17280 z^2 w^2 + 27000 x^3 u + 27000 y^3 u + 46656 y z^2 u + 46656 x w^2 u + 15552 x y u^2 + 3888 z w u^2 - 8748 u^4 - 32400 x^2 w - 32400 y^2 z - 13824 z^3 - 13824 w^3 - 77760 x z u - 77760 y w u + 38880 x y + 62208 z w + 34992 u^2 - 46656 (* Check: $\disc_6^+(x,y,z,w,u)$ is a discriminant. *) (* Calculate the coefficients of S51, S15, S42, S24, S33 in the polynomial G164[s,1,0,s] + b G264[s,1,0,s] + c G364[s,1,0,s] + d G464[s,1,0,s]. *) Factor[Disc6p[2 b, 2 c s^3, ((1 - 2 s^6)/s^4 - 3 b s + c + d), ((s^8 - 2 s^2)/s^4 + b s^3 - 3 c s^2 + d s^2), -2 d s]] Factor[Disc6p[p,p,q,q,r]] (*--------------------------------------------------------------------------*) (* Omittes part of Lemma 5.8. *) 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) F2[a_,b_,c_,s_] := 2 S41[a,b,c] + s^3 S14[a,b,c] - 3s S32[a,b,c]-(s^3-3s+2)US2[a,b,c] (* Check: $F_{2,s}(s,1,0) = 0$ *) Factor[F2[s,1,0,s]] (* Check: $s^3 F_{2,s}(a, b, c, 1/s) = 2p s^3 - 3q s^2 + r, p := S_{4,1} - US_2, q := S_{3,2} - US_2, r := S_{1,4} - US_2. *) Factor[s^3 F2[a,b,c,1/s] - (2s^3 (S41[a,b,c] - US2[a,b,c]) - 3 s^2 (S32[a,b,c] - US2[a,b,c]) + (S14[a,b,c] - US2[a,b,c]))] (* Check: p^2r - q^3 = (S_{4,1}-US_2)^2 (S_{1,4}-US_2) - (S_{3,2}-US_2)^3 = U (S_2 - S_{1,1}) \{ S_{9,1} - S_{7,3} + S_{6,4} + 2S_{4,6} - US_7 + 2 US_{6,1} - US_{1,6} - 3 US_{5,2} - 2US_{2,5} + US_{4,3} - 2US_{3,4} - U^2S_4 - U^2S_{3,1} + 5U^2S_{1,3} - 2U^2S_{2,2} + 2U^3S_1 \} *) S91[a_, b_, c_] := a^9 b + b^9 c + c^9 a S73[a_, b_, c_] := a^7 b^3 + b^7 c^3 + c^7 a^3 S64[a_, b_, c_] := a^6 b^4 + b^6 c^4 + c^6 a^4 S46[a_, b_, c_] := a^4 b^6 + b^4 c^6 + c^4 a^6 US7[a_, b_, c_] := a b c(a^7+b^7+c^7) US61[a_, b_, c_] := a b c(a^6 b+b^6 c+c^6 a) US16[a_, b_, c_] := a b c(a b^6+b c^6+c a^6) US52[a_, b_, c_] := a b c(a^5 b^2+b^5 c^2+c^5 a^2) US25[a_, b_, c_] := a b c(a^2 b^5+b^2 c^5+c^2 a^5) US43[a_, b_, c_] := a b c(a^4 b^3+b^4 c^3+c^4 a^3) US34[a_, b_, c_] := a b c(a^3 b^4+b^3 c^4+c^3 a^4) U2S4[a_, b_, c_] := a^2 b^2 c^2(a^4+b^4+c^4) U2S31[a_, b_, c_] := a^2 b^2 c^2(a^3 b+b^3 c+c^3 a) U2S13[a_, b_, c_] := a^2 b^2 c^2(a b^3+b c^3+c a^3) U2S22[a_, b_, c_] := a^2 b^2 c^2(a^2 b^2+b^2 c^2+c^2 a^2) U3S1[a_, b_, c_] := a^3 b^3 c^3(a+b+c) Factor[((S41[a,b,c]-US2[a,b,c])^2 (S14[a,b,c]-US2[a,b,c]) - (S32[a,b,c]-US2[a,b,c])^3) - a b c(S2[a,b,c] - S11[a,b,c]) (S91[a,b,c] - S73[a,b,c] + S64[a,b,c] + 2S46[a,b,c] - US7[a,b,c] + 2 US61[a,b,c] - US16[a,b,c] - 3 US52[a,b,c]- 2US25[a,b,c] + US43[a,b,c] - 2US34[a,b,c] - U2S4[a,b,c] - U2S31[a,b,c] + 5U2S13[a,b,c] - 2U2S22[a,b,c] + 2U3S1[a,b,c])] (* Check: $g(a, 1-k(1-a), 1) = (1-a)^4 \sum_{i=0}^6 \varphi_i(k) a^i$ *) (* G71[a,b,c] is g(a,b,c) *) G71[a_,b_,c_] := S91[a,b,c] - S73[a,b,c] + S64[a,b,c] + 2S46[a,b,c] - US7[a,b,c] + 2 US61[a,b,c] - US16[a,b,c] - 3 US52[a,b,c]- 2US25[a,b,c] + US43[a,b,c] - 2US34[a,b,c] - U2S4[a,b,c] - U2S31[a,b,c] + 5U2S13[a,b,c] - 2U2S22[a,b,c] + 2U3S1[a,b,c] phi0[k_] := (1-k)^4(3 - 4 k + 8 k^2 - 9 k^3 + 5 k^4 - k^5) phi1[k_] := 6 - 24 k + 44 k^2 - 31 k^3 - 42 k^4 + 142 k^5 - 173 k^6 + 111 k^7 - 37 k^8 + 5 k^9 phi2[k_] := 7 - 13 k + 2 k^2 + 37 k^3 - 57 k^4 - 3 k^5 + 104 k^6 - 122 k^7 + 58 k^8 - 10 k^9 phi3[k_] := 6 - 5 k - 3 k^2 + 17 k^3 - 3 k^4 - 21 k^5 - 3 k^6 + 50 k^7 - 42 k^8 + 10 k^9 phi4[k_] := 3 + 3 k - 6 k^2 + 3 k^3 + 18 k^4 - 17 k^5 - 2 k^6 - 3 k^7 + 13 k^8 - 5 k^9 phi5[k_] := 1 + 2 k - k^2 + k^4 + 10 k^5 - 6 k^6 - k^7 - k^8 + k^9 phi6[k_] := k (1 - k^2 + k^3 + 2 k^5) Factor[G71[a, 1-k(1-a), 1] - (1-a)^4 (phi6[k] a^6 + phi5[k] a^5 + phi4[k] a^4 + phi3[k] a^3 + phi2[k] a^2 + phi1[k] a + phi0[k])] Plot[phi0[k], {k, 0, 1}] Plot[phi1[k], {k, 0, 1}] Plot[phi2[k], {k, 0, 1}] Plot[phi3[k], {k, 0, 1}] Plot[phi4[k], {k, 0, 1}] Plot[phi5[k], {k, 0, 1}] Plot[phi6[k], {k, 0, 1}] (* Check: $g(1-k(1-b), b, 1) = (1-b)^4 \sum_{i=0}^6 \psi_i(k) b^i$ *) psi0[k_] := (1-k)(3 - 11 k + 22 k^2 - 21 k^3 + 10 k^4 - 2 k^5) psi1[k_] := 6 - 12 k + 2 k^2 + 23 k^3 - 18 k^4 - 23 k^5 + 43 k^6 - 27 k^7 + 8 k^8 - k^9 psi2[k_] := 7 - 13 k + 32 k^2 - 68 k^3 + 78 k^4 - 6 k^5 - 79 k^6 + 79 k^7 - 32 k^8 + 5 k^9 psi3[k_] := 6 - 9 k + 9 k^2 + 27 k^3 - 83 k^4 + 81 k^5 + 11 k^6 - 74 k^7 + 48 k^8 - 10 k^9 psi4[k_] := 3 - 3 k + 3 k^2 + 3 k^3 + 18 k^4 - 47 k^5 + 34 k^6 + 18 k^7 - 32 k^8 + 10 k^9 psi5[k_] := 1 - k - k^2 + 6 k^3 - 2 k^4 + 7 k^5 - 12 k^6 + 5 k^7 + 8 k^8 - 5 k^9 psi6[k_] := k^4 (2 + k^2 - k^3 + k^5) Factor[G71[1-k(1-b), b, 1] - (1-b)^4 (psi6[k] b^6 + psi5[k] b^5 + psi4[k] b^4 + psi3[k] b^3 + psi2[k] b^2 + psi1[k] b + psi0[k])] Plot[psi0[k], {k, 0, 1}] Plot[psi1[k], {k, 0, 1}] Plot[psi2[k], {k, 0, 1}] Plot[psi3[k], {k, 0, 1}] Plot[psi4[k], {k, 0, 1}] Plot[psi5[k], {k, 0, 1}] Plot[psi6[k], {k, 0, 1}] (*---------------------------------------------------------------------------*) (* Omittes part of Lemma 6.6. *) 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) G2[a_,b_,c_,s_]:= 2 S51[a,b,c] - 3 s S42[a,b,c] + s^3 S24[a,b,c] - 3 (s-1)^2(s+2) U2[a,b,c] (* Check: $G_{2,s}(s,1,0) = 0$ *) Factor[G2[s,1,0,s]] (* Check: $s^3 G_{2,1/s}(a,b,c) = 2 s^3 p - 3 s^2 q + r$ $p := S_{5,1}-3U^2$, $q := S_{4,2} - 3U^2$, $r := S_{2,4}-3U^2$ *) Factor[s^3 G2[a,b,c,1/s] - (2 s^3 (S51[a,b,c]-3U2[a,b,c]) - 3 s^2 (S42[a,b,c] - 3U2[a,b,c]) + (S24[a,b,c]-3U2[a,b,c]))] (* Check: (p^2r - q^3)/U = 2S_{6,9} + US_{12} - 3 US_{10,2} + 9 US_{8,4} - 6 U S_{7,5} - 3US_{6,6} + 2 U^2 S_{6,3} - 6U^2 S_{4,5} - 6U^2 S_{1,8} - 2U^3 S_6 + 18 U^3 S_{5,1} - 27 U^3 S_{4,2} + 27 U^3 S_{2,4} + 2U^4S_3 - 6 U^4S_{1,2} - 6U^5. *) S69[a_, b_, c_] := a^6 b^9 + b^6 c^9 + c^6 a^9 USc[a_, b_, c_] := a b c(a^(12)+b^(12)+c^(12)) USa2[a_, b_, c_] := a b c(a^(10) b^2+b^(10) c^2+c^(10) a^2) US84[a_, b_, c_] := a b c(a^8 b^4+b^8 c^4+c^8 a^4) US75[a_, b_, c_] := a b c(a^7 b^5+b^7 c^5+c^7 a^5) US66[a_, b_, c_] := a b c(a^6 b^6+b^6 c^6+c^6 a^6) U2S63[a_, b_, c_] := a^2 b^2 c^2(a^6 b^3+b^6 c^3+c^6 a^3) U2S45[a_, b_, c_] := a^2 b^2 c^2(a^4 b^5+b^4 c^5+c^4 a^5) U2S18[a_, b_, c_] := a^2 b^2 c^2(a b^8+b c^8+c a^8) U3S6[a_, b_, c_] := a^3 b^3 c^3(a^6+b^6+c^6) U3S51[a_, b_, c_] := a^3 b^3 c^3(a^5 b+b^5 c+c^5 a) U3S42[a_, b_, c_] := a^3 b^3 c^3(a^4 b^2+b^4 c^2+c^4 a^2) U3S24[a_, b_, c_] := a^3 b^3 c^3(a^2 b^4+b^2 c^4+c^2 a^4) U4S3[a_, b_, c_] := a^4 b^4 c^4(a^3+b^3+c^3) U4S12[a_, b_, c_] := a^4 b^4 c^4(a b^2+b c^2+c a^2) U5[a_, b_, c_] := a^5 b^5 c^5 G72[a_,b_,c_] := 2 S69[a,b,c] + USc[a,b,c] - 3 USa2[a,b,c] + 9 US84[a,b,c] - 6 US75[a,b,c] - 3 US66[a,b,c] + 2 U2S63[a,b,c] - 6 U2S45[a,b,c] - 6 U2S18[a,b,c] - 2 U3S6[a,b,c] + 18 U3S51[a,b,c] - 27 U3S42[a,b,c] + 27 U3S24[a,b,c] + 2 U4S3[a,b,c] - 6 U4S12[a,b,c] - 6 U5[a,b,c] Factor[((S51[a,b,c]-3U2[a,b,c])^2 (S24[a,b,c]-3U2[a,b,c]) - (S42[a,b,c] - 3U2[a,b,c])^3) - a b c G72[a,b,c]] (* Check: $g(1 - k (1 - b), b, 1) = (1-b)^6 \sum_{i=0}^9 \varphi_i(k) b^i$ *) phi0[k_] := 2 (1 - k)^9 phi1[k_] := (1-k)(11 - 72 k + 222 k^2 - 430 k^3 + 639 k^4 - 810 k^5 + 879 k^6 - 762 k^7 + 489 k^8 - 220 k^9 + 66 k^(10) - 12 k^(11) + k^(12)) phi2[k_] := 26 - 143 k + 368 k^2 - 542 k^3 + 426 k^4 + 187 k^5 - 1302 k^6 + 2439 k^7 - 2826 k^8 + 2199 k^9 - 1150 k^(10) + 390 k^(11) - 78 k^(12) + 7 k^(13) phi3[k_] := 36 - 122 k + 152 k^2 + 80 k^3 - 631 k^4 + 1285 k^5 - 1365 k^6 + 309 k^7 + 1314 k^8 - 2156 k^9 + 1707 k^(10) - 777 k^(11) + 195 k^(12) - 21 k^(13) phi4[k_] := 32 - 50 k - 28 k^2 + 248 k^3 - 446 k^4 + 271 k^5 + 512 k^6 - 1181 k^7 + 774 k^8 + 394 k^9 - 1048 k^(10) + 765 k^(11) - 260 k^(12) + 35 k^(13) phi5[k_] := 18 + 2 k - 48 k^2 + 96 k^3 + 32 k^4 - 402 k^5 + 623 k^6 - 167 k^7 - 466 k^8 + 432 k^9 + 112 k^(10) - 360 k^(11) + 195 k^(12) - 35 k^(13) phi6[k_] := 6 + 12 k - 18 k^2 + 20 k^3 + 60 k^4 - 60 k^5 - 168 k^6 + 361 k^7 - 154 k^8 - 140 k^9 + 126 k^(10) + 48 k^(11) - 78 k^(12) + 21 k^(13) phi7[k_] := 1 + 5 k - 6 k^3 + 30 k^4 + 12 k^5 - 27 k^6 - 45 k^7 + 95 k^8 - 27 k^9 - 33 k^(10) + 15 k^(11) + 13 k^(12) - 7 k^(13) phi8[k_] := k (1 + 12 k^4 - 3 k^6 - 6 k^7 + 9 k^8 - 3 k^(10) + k^(12)) phi9[k_] := 2 k^6 Factor[G72[1-k(1-b), b, 1] - (1-b)^6 (phi9[k] b^9 + phi8[k] b^8 + phi7[k] b^7 + phi6[k] b^6 + phi5[k] b^5 + phi4[k] b^4 + phi3[k] b^3 + phi2[k] b^2 + phi1[k] b + phi0[k])] Plot[phi0[k], {k, 0, 1}] Plot[phi1[k], {k, 0, 1}] Plot[phi2[k], {k, 0, 1}] Plot[phi3[k], {k, 0, 1}] Plot[phi4[k], {k, 0, 1}] Plot[phi5[k], {k, 0, 1}] Plot[phi6[k], {k, 0, 1}] Plot[phi7[k], {k, 0, 1}] Plot[phi8[k], {k, 0, 1}] Plot[phi9[k], {k, 0, 1}] (* Check: $q(a, 1 - k (1 - a), 1) = (1 - a)^6 \sum_{i=0}^9 \psi_i(k) a^i$ *) psi0[k_] := 2 (1 - k)^6 psi1[k_] := (1-k)(11 - 42 k + 84 k^2 - 142 k^3 + 252 k^4 - 396 k^5 + 501 k^6 - 498 k^7 + 369 k^8 - 190 k^9 + 63 k^(10) - 12 k^(11) + k^(12)) psi2[k_] := 26 - 107 k + 272 k^2 - 544 k^3 + 808 k^4 - 724 k^5 + 50 k^6 + 983 k^7 - 1732 k^8 + 1677 k^9 - 1012 k^(10) + 375 k^(11) - 78 k^(12) + 7 k^(13) psi3[k_] := 36 - 128 k + 311 k^2 - 437 k^3 + 160 k^4 + 572 k^5 - 1113 k^6 + 691 k^7 + 593 k^8 - 1615 k^9 + 1512 k^(10) - 750 k^(11) + 195 k^(12) - 21 k^(13) psi4[k_] := 32 - 86 k + 134 k^2 + 59 k^3 - 446 k^4 + 536 k^5 + 14 k^6 - 677 k^7 + 542 k^8 + 373 k^9 - 988 k^(10) + 750 k^(11) - 260 k^(12) + 35 k^(13) psi5[k_] := 18 - 22 k - 30 k^2 + 206 k^3 - 227 k^4 + 11 k^5 + 183 k^6 - 25 k^7 - 270 k^8 + 186 k^9 + 217 k^(10) - 375 k^(11) + 195 k^(12) - 35 k^(13) psi6[k_] := 6 + 6 k - 42 k^2 + 70 k^3 + 28 k^4 - 115 k^5 + 126 k^6 - 39 k^7 + 18 k^8 - 64 k^9 + 24 k^(10) + 75 k^(11) - 78 k^(12) + 21 k^(13) psi7[k_] := 1 + 5 k - 9 k^2 - 9 k^3 + 45 k^4 - 25 k^5 - 21 k^6 + 69 k^7 - 36 k^8 + 6 k^9 - 6 k^(10) + 13 k^(12) - 7 k^(13) psi8[k_] := k (1 - 3 k^2 + 9 k^4 - 6 k^5 - 3 k^6 + 18 k^7 - 6 k^8 + k^(12)) psi9[k_] := 2 k^9 Factor[G72[a, 1-k(1-a), 1] - (1-a)^6 (psi9[k] a^9 + psi8[k] a^8 + psi7[k] a^7 + psi6[k] a^6 + psi5[k] a^5 + psi4[k] a^4 + psi3[k] a^3 + psi2[k] a^2 + psi1[k] a + psi0[k])] Plot[psi0[k], {k, 0, 1}] Plot[psi1[k], {k, 0, 1}] Plot[psi2[k], {k, 0, 1}] Plot[psi3[k], {k, 0, 1}] Plot[psi4[k], {k, 0, 1}] Plot[psi5[k], {k, 0, 1}] Plot[psi6[k], {k, 0, 1}] Plot[psi7[k], {k, 0, 1}] Plot[psi8[k], {k, 0, 1}] Plot[psi9[k], {k, 0, 1}] (*-------------------------------------------------------------------------------*) (* Theorem 8.1. *) 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) F181[a_,b_,c_,s_] := 3s^4 S5[a,b,c] - (4s^5 - 1) S41[a,b,c] + (s^8 - 4s^3) S14[a,b,c] - (s^8 - 4s^5 + 3s^4 - 4s^3 + 1) US2[a,b,c] (* Chack: $F_{1,s}(s,1,0) = 0$ *) Factor[F181[s,1,0,s]] (* Check: $s^8 F_{1,1/s}(a$, $b$, $c) = G_s(a$, $b$, $c)$. *) G81[a_,b_,c_,s_] := (s^8 - 4s^3)(S41[a,b,c] - US2[a,b,c]) + (1-4s^5)(S14[a,b,c]-US2[a,b,c]) + 3s^4(S5[a,b,c] - US2[a,b,c]) Factor[s^8 F181[a,b,c,1/s] - G81[a,b,c,s]] (* Check: $G_s(a, 1-k(1-a), 1) = (1-a)^2 \sum_{i=0}^3 C_i(k,s) a^i$. *) C081[k_,s_] := (1 - s + k s)^2 (1 - k + 2 s - 4 k s + 2 k^2 s + 3 s^2 - 9 k s^2 + 9 k^2 s^2 - 3 k^3 s^2 + 3 s^4 + 2 s^5 - 2 k s^5 + s^6 - 2 k s^6 + k^2 s^6) C181[k_,s_] := 1 - k + 3 k^2 - 3 k^3 + k^4 - 4 s^3 + 12 k^2 s^3 - 20 k^3 s^3 + 8 k^4 s^3 + 6 s^4 - 3 k s^4 - 9 k^2 s^4 + 33 k^3 s^4 - 30 k^4 s^4 + 9 k^5 s^4 - 4 s^5 + 4 k s^5 - 12 k^2 s^5 + 12 k^3 s^5 - 4 k^4 s^5 + s^8 - 3 k^2 s^8 + 5 k^3 s^8 - 2 k^4 s^8 C281[k_,s_] := 1 - k + 3 k^3 - 2 k^4 - 4 s^3 + 4 k^3 s^3 - 4 k^4 s^3 + 6 s^4 - 3 k s^4 - 3 k^3 s^4 + 15 k^4 s^4 - 9 k^5 s^4 - 4 s^5 + 4 k s^5 - 12 k^3 s^5 + 8 k^4 s^5 + s^8 - k^3 s^8 + k^4 s^8 C381[k_,s_] := (k - s)^2 (k^2 + 2 k s + 3 s^2 + 3 k^3 s^4 + 2 k^2 s^5 + k s^6) Factor[G81[a, 1-k(1-a), 1, s] - (1-a)^2(C381[k,s] a^3 + C281[k,s] a^2 + C181[k,s] a + C081[k,s])] (* Check: $\sum_{i=0}^3 C_i(k,s) a^i = \sum_{i=0}^2 A_i(k,s) a^i(1-a) + A_3(k,s) a^3$ *) A081[k_,s_] := C081[k,s] A181[k_,s_] := C081[k,s] + C181[k,s] A281[k_,s_] := C081[k,s] + C181[k,s] + C281[k,s] A381[k_,s_] := C081[k,s] + C181[k,s] + C281[k,s] + C381[k,s] Factor[(C381[k,s] a^3 + C281[k,s] a^2 + C181[k,s] a + C081[k,s]) - ( A081[k,s] (1-a) + A181[k,s] a(1-a) + A281[k,s] a^2(1-a) + A381[k,s] a^3)] (*-----------------------*) (* Theorem 8.1. Case 1 *) (* Case 1-1 *) (* Check: ${1 \over (1-s)^2} A_1((1 - s) x, s) = \sum_{i=0}^4 a_i(s) (1-x)x^i + a_5(s)x^5$ *) a0[s_] := 2 + 4 s + 6 s^2 + 6 s^4 + 4 s^5 + 2 s^6 a1[s_] := 2 s (1 + 2 s + 7 s^2 + s^3 + 4 s^4 + 3 s^5 + 2 s^6) a2[s_] := 3 + 2 s + 4 s^2 + 2 s^3 + 23 s^4 - 4 s^5 + 6 s^6 + 4 s^7 + 3 s^8 a3[s_] := s(5 + 4 s - 2 s^2 + 30 s^3 + 5 s^4 - 6 s^5 + 4 s^6 + 4 s^7 - s^8) a4[s_] := 1 + 3 s + 5 s^2 + 2 s^3 + 7 s^4 + 35 s^5 - 13 s^6 + 3 s^8 + s^9 - s^(10) a5[s_] := 1 + 3 s + 5 s^2 + 2 s^3 + 13 s^4 + 17 s^5 + 5 s^6 - 6 s^7 + 3 s^8 + s^9 - s^(10) Factor[(1/(1-s)^2) A181[(1-s)x, s] - (a0[s] (1-x) + a1[s] (1-x) x + a2[s] (1-x) x^2 + a3[s] (1-x) x^3 + a4[s] (1-x) x^4 + a5[s] x^5)] Plot[a0[s], {s, 0, 1}] Plot[a1[s], {s, 0, 1}] Plot[a2[s], {s, 0, 1}] Plot[a3[s], {s, 0, 1}] Plot[a4[s], {s, 0, 1}] Plot[a5[s], {s, 0, 1}] (* Theorem 8.1. Case 1-2 *) (* Check: ${1 \over (1-m)^2} A_1(1-m, 1-x+mx) = \sum_{i=0}^9 b_i(x) (1-m) m^i + b_{10} m^{10}$ *) b0[x_] := 3 - 12 x + 30 x^2 - 48 x^3 + 59 x^4 - 52 x^5 + 28 x^6 - 8 x^7 + x^8 b1[x_] := 6 + 4 x - 44 x^2 + 104 x^3 - 152 x^4 + 164 x^5 - 112 x^6 + 40 x^7 - 6 x^8 b2[x_] := 9 + 20 x - 44 x^2 + 93 x^4 - 172 x^5 + 168 x^6 - 80 x^7 + 15 x^8 b3[x_] := 3 + 40 x + 26 x^2 - 184 x^3 + 227 x^4 - 112 x^5 - 28 x^6 + 56 x^7 - 17 x^8 b4[x_] := 3 + 16 x + 60 x^2 + 32 x^3 - 302 x^4 + 404 x^5 - 252 x^6 + 64 x^7 - x^8 b5[x_] := 3 + 16 x + 24 x^2 + 24 x^3 + 69 x^4 - 292 x^5 + 336 x^6 - 168 x^7 + 29 x^8 b6[x_] := 3 + 16 x + 24 x^2 + 6 x^4 + 60 x^5 - 168 x^6 + 144 x^7 - 39 x^8 b7[x_] := 3 + 16 x + 24 x^2 + 28 x^6 - 56 x^7 + 25 x^8 b8[x_] := 3 + 16 x + 24 x^2 + 8 x^7 - 8 x^8 b9[x_] := 3 + 16 x + 24 x^2 + x^8 b10[x_] := 3 + 16 x + 24 x^2 Factor[(1/(1-m)^2) A181[1-m, 1-x+m x] - (b0[x] (1-m) + b1[x] (1-m) m + b2[x] (1-m) m^2 + b3[x] (1-m) m^3 + b4[x] (1-m) m^4 + b5[x] (1-m) m^5 + b6[x] (1-m) m^6 + b7[x] (1-m) m^7 + b8[x] (1-m) m^8 + b9[x] (1-m) m^9 + b10[x] m^(10))] Plot[b0[s], {s, 0, 1}] Plot[b1[s], {s, 0, 1}] Plot[b2[s], {s, 0, 1}] Plot[b3[s], {s, 0, 1}] Plot[b4[s], {s, 0, 1}] Plot[b5[s], {s, 0, 1}] Plot[b6[s], {s, 0, 1}] Plot[b7[s], {s, 0, 1}] Plot[b8[s], {s, 0, 1}] Plot[b9[s], {s, 0, 1}] Plot[b10[s], {s, 0, 1}] (* Theorem 8.1. Case 1-3 *) (* Check: ${1 \over (1 - s)^2} A_2((1 - s) x, s) = \sum_{i=0}^4 c_i(s) (1-x) x^i + c_5(s) x^i$ *) c0[s_] := 3 (1 + 2 s + 3 s^2 + 3 s^4 + 2 s^5 + s^6) c1[s_] := s (3 + 6 s + 13 s^2 + s^3 + 10 s^4 + 7 s^5 + 4 s^6) c2[s_] := 3 + 3 s + 6 s^2 + s^3 + 22 s^4 - 2 s^5 + 7 s^6 + 4 s^7 + 3 s^8 c3[s_] := c2[s] c4[s_] := 2 + 5 s + 5 s^2 + s^3 + 22 s^4 + 2 s^5 - s^6 + 8 s^7 + 3 s^8 c5[s_] := 2 + 5 s + 5 s^2 + s^3 + 19 s^4 + 11 s^5 - 10 s^6 + 11 s^7 + 3 s^8 Factor[(1/(1-s)^2) A281[(1-s)x, s] - (c0[s](1-x) + c1[s] (1-x) x + c2[s] (1-x) x^2 + c3[s] (1-x) x^3 + c4[s] (1-x) x^4 + c5[s] x^5)] Plot[c0[s], {s, 0, 1}] Plot[c1[s], {s, 0, 1}] Plot[c2[s], {s, 0, 1}] Plot[c3[s], {s, 0, 1}] Plot[c4[s], {s, 0, 1}] Plot[c5[s], {s, 0, 1}] (* Theorem 8.1. Case 1-4 *) (* Check: ${1 \over (1-m)^2} A_2(1-m, 1-x+mx) = \sum_{i=0}^7 d_i(x) (1-m)m^i + d_8(x) m^8$ *) d0[x_] := 3 - 12 x + 42 x^2 - 84 x^3 + 115 x^4 - 104 x^5 + 56 x^6 - 16 x^7 + 2 x^8 d1[x_] := 6 + 12 x - 66 x^2 + 168 x^3 - 281 x^4 + 324 x^5 - 224 x^6 + 80 x^7 - 12 x^8 d2[x_] := 24 x + 48 x^2 - 240 x^3 + 427 x^4 - 516 x^5 + 420 x^6 - 184 x^7 + 33 x^8 d3[x_] := 3 - 4 x + 68 x^2 + 40 x^3 - 311 x^4 + 492 x^5 - 476 x^6 + 256 x^7 - 55 x^8 d4[x_] := 3 + 8 x + 18 x^2 + 20 x^3 + 55 x^4 - 216 x^5 + 308 x^6 - 224 x^7 + 60 x^8 d5[x_] := 3 + 8 x + 36 x^2 - 12 x^3 - 2 x^4 + 24 x^5 - 84 x^6 + 112 x^7 - 42 x^8 d6[x_] := 3 + 8 x + 36 x^2 - 3 x^4 - 4 x^5 - 24 x^7 + 17 x^8 d7[x_] := 3 + 8 x + 36 x^2 - 3 x^8 d8[x_] := 3 + 8 x + 36 x^2 Factor[(1/(1-m)^2) A281[1-m, 1-x+m x] - (d0[x] (1-m) + d1[x] (1-m) m + d2[x] (1-m) m^2 + d3[x] (1-m) m^3 + d4[x] (1-m) m^4 + d5[x] (1-m) m^5 + d6[x] (1-m) m^6 + d7[x] (1-m) m^7 + d8[x] m^8)] Plot[d0[s], {s, 0, 1}] Plot[d1[s], {s, 0, 1}] Plot[d2[s], {s, 0, 1}] Plot[d3[s], {s, 0, 1}] Plot[d4[s], {s, 0, 1}] Plot[d5[s], {s, 0, 1}] Plot[d6[s], {s, 0, 1}] Plot[d7[s], {s, 0, 1}] Plot[d8[s], {s, 0, 1}] (*-----------------------*) (* Theorem 8.1. Case 2 *) (* Check: $G_s(1-k(1-b), b, 1) = (1-b)^2 (\sum_{i=0}^2 B_i(k,s) b^i(1-b) + B_3(k,s) b^3)$ *) (* B_0(1-k,s) = (k - s)^2 (k^2 + 2 k s + 3 s^2 + 3 k^3 s^4 + 2 k^2 s^5 + k s^6) *) B081[k_,s_] := (1-k - s)^2 ((1-k)^2 + 2 (1-k) s + 3 s^2 + 3 (1-k)^3 s^4 + 2 (1-k)^2 s^5 + (1-k) s^6) B181[k_,s_] := 2 - 4 k + 3 k^2 + k^3 - k^4 - 8 s^3 + 8 k s^3 - 12 k^2 s^3 + 12 k^3 s^3 - 4 k^4 s^3 + 12 s^4 - 18 k s^4 + 21 k^2 s^4 + 3 k^3 s^4 - 15 k^4 s^4 + 6 k^5 s^4 - 8 s^5 + 16 k s^5 - 12 k^2 s^5 - 4 k^3 s^5 + 4 k^4 s^5 + 2 s^8 - 2 k s^8 + 3 k^2 s^8 - 3 k^3 s^8 + k^4 s^8 B281[k_,s_] := 3 - 4 k + 3 k^2 - 12 s^3 + 12 k s^3 - 12 k^2 s^3 + 4 k^4 s^3 + 18 s^4 - 21 k s^4 + 21 k^2 s^4 - 3 k^5 s^4 - 12 s^5 + 16 k s^5 - 12 k^2 s^5 + 3 s^8 - 3 k s^8 + 3 k^2 s^8 - k^4 s^8 B381[k_,s_] := B281[k,s] + (1-k s)^2 (k + 2 k^2 s + 3 k^3 s^2 + 3 s^4 + 2 k s^5 + k^2 s^6) Factor[G81[1-k(1-b), b, 1, s] - (1-b)^2(B081[k,s] (1-b) + B181[k,s] b(1-b) + B281[k,s] b^2(1-b) + B381[k,s] b^3)] (* Theorem 8.1. Case 2-1 *) (* Check: ${1 \over (1-s)^2} B_1((1 - s) x, s) = \sum_{i=0}^{10} e_i(x) s^i$ *) e0[x_] := 2 - 4 x + 3 x^2 + x^3 - x^4 e1[x_] := 4 - 4 x - x^3 + 2 x^4 e2[x_] := 6 - 4 x - x^4 e3[x_] := 4 x - 12 x^2 + 12 x^3 - 4 x^4 e4[x_] := 6 - 14 x + 21 x^2 - 9 x^3 - 7 x^4 + 6 x^5 e5[x_] := 4 + 2 x - 12 x^2 - 7 x^3 + 30 x^4 - 18 x^5 e6[x_] := 2 + 2 x + 4 x^3 - 23 x^4 + 18 x^5 e7[x_] := 2 x + 4 x^4 - 6 x^5 e8[x_] := 3 x^2 - 3 x^3 + x^4 e9[x_] := 3 x^3 - 2 x^4 e10[x_] := x^4 Factor[(1/(1-s)^2) B181[(1-s)x, s] - (e0[x] + e1[x] s + e2[x] s^2 + e3[x] s^3 + e4[x] s^4 + e5[x] s^5 + e6[x] s^6 + e7[x] s^7 + e8[x] s^8 + e9[x] s^9 + e10[x] s^(10))] Plot[e0[s], {s, 0, 1}] Plot[e1[s], {s, 0, 1}] Plot[e2[s], {s, 0, 1}] Plot[e3[s], {s, 0, 1}] Plot[e4[s], {s, 0, 1}] Plot[e5[s], {s, 0, 1}] Plot[e6[s], {s, 0, 1}] Plot[e7[s], {s, 0, 1}] Plot[e8[s], {s, 0, 1}] Plot[e9[s], {s, 0, 1}] Plot[e10[s], {s, 0, 1}] Plot[e5[s]+e0[s], {s, 0, 1}] (* Theorem 8.1. Case 2-2 *) (* Cjeck: ${1 \over (1-m)^2} B_1((1-m), 1-(1-m)x) = \sum_{i=0}^9 f_i(x) (1-m) m^i + f_{10}(x) m^{10}$ *) f0[x_] := 3 - 12 x + 30 x^2 - 48 x^3 + 59 x^4 - 52 x^5 + 28 x^6 - 8 x^7 + x^8 f1[x_] := 6 - 28 x + 68 x^2 - 80 x^3 + 28 x^4 + 44 x^5 - 56 x^6 + 24 x^7 - 4 x^8 f2[x_] := 9 - 44 x + 68 x^2 - 87 x^4 + 68 x^5 - 16 x^7 + 5 x^8 f3[x_] := 3 + 8 x - 86 x^2 + 184 x^3 - 133 x^4 + 8 x^5 + 28 x^6 - 8 x^7 - x^8 f4[x_] := 3 - 16 x + 60 x^2 - 152 x^3 + 238 x^4 - 196 x^5 + 84 x^6 - 16 x^7 + x^8 f5[x_] := 3 - 16 x + 24 x^2 + 24 x^3 - 111 x^4 + 188 x^5 - 168 x^6 + 72 x^7 - 11 x^8 f6[x_] := 3 - 16 x + 24 x^2 + 6 x^4 - 60 x^5 + 112 x^6 - 80 x^7 + 19 x^8 f7[x_] := 3 - 16 x + 24 x^2 - 28 x^6 + 40 x^7 - 15 x^8 f8[x_] := 3 - 16 x + 24 x^2 - 8 x^7 + 6 x^8 f9[x_] := 3 - 16 x + 24 x^2 - x^8 f10[x_] := 3 - 16 x + 24 x^2 Factor[(1/(1-m)^2) B181[(1-m), 1-(1-m)x] - (f0[x] (1-m) + f1[x] (1-m) m + f2[x] (1-m) m^2 + f3[x] (1-m) m^3 + f4[x] (1-m) m^4 + f5[x] (1-m) m^5 + f6[x] (1-m) m^6 + f7[x] (1-m) m^7 + f8[x] (1-m) m^8 + f9[x] (1-m) m^9 + f10[x] m^(10))] Plot[f0[s], {s, 0, 1}] Plot[f1[s], {s, 0, 1}] Plot[f2[s], {s, 0, 1}] Plot[f3[s], {s, 0, 1}] Plot[f4[s], {s, 0, 1}] Plot[f5[s], {s, 0, 1}] Plot[f6[s], {s, 0, 1}] Plot[f7[s], {s, 0, 1}] Plot[f8[s], {s, 0, 1}] Plot[f9[s], {s, 0, 1}] Plot[f10[s], {s, 0, 1}] (* Theorem 8.1. Case 2-3 *) (* ${1 \over (1 - s)^2} B_2((1-s) x, s) = \sum_{i=0}^{10} g_i(x) s^i$ *) g0[x_] := 3 - 4 x + 3 x^2 g1[x_] := 6 - 4 x g2[x_] := 9 - 4 x g3[x_] := 8 x - 12 x^2 + 4 x^4 g4[x_] := 9 - 13 x + 21 x^2 - 8 x^4 - 3 x^5 g5[x_] := 6 + 3 x - 12 x^2 + 4 x^4 + 9 x^5 g6[x_] := 3 + 3 x - 9 x^5 g7[x_] := 3 x + 3 x^5 g8[x_] := 3 x^2 - x^4 g9[x_] := 2 x^4 g10[x_] := - x^4 Factor[(1/(1-s)^2) B281[(1-s)x, s] - (g0[x] + g1[x] s + g2[x] s^2 + g3[x] s^3 + g4[x] s^4 + g5[x] s^5 + g6[x] s^6 + g7[x] s^7 + g8[x] s^8 + g9[x] s^9 + g10[x] s^(10))] Plot[g0[s], {s, 0, 1}] Plot[g1[s], {s, 0, 1}] Plot[g2[s], {s, 0, 1}] Plot[g3[s], {s, 0, 1}] Plot[g4[s], {s, 0, 1}] Plot[g5[s], {s, 0, 1}] Plot[g6[s], {s, 0, 1}] Plot[g7[s], {s, 0, 1}] Plot[g8[s], {s, 0, 1}] Plot[g9[s], {s, 0, 1}] Plot[g10[s], {s, 0, 1}] Plot[g5[s]+g6[s], {s, 0, 1}] Plot[g9[s]+g10[s], {s, 0, 1}] (* Theorem 8.1. Case 2-4 *) (* Check: ${1 \over (1-m)^2} B_2(1-m, 1-x+mx) = \sum_{i=0}^9 h_i(x) (1-m)m^i + h_{10}(x) m^{10}$. *) h0[x_] := 3 - 12 x + 42 x^2 - 84 x^3 + 115 x^4 - 104 x^5 + 56 x^6 - 16 x^7 + 2 x^8 h1[x_] := 6 - 36 x + 102 x^2 - 108 x^3 - 11 x^4 + 144 x^5 - 140 x^6 + 56 x^7 - 9 x^8 h2[x_] := x (24 - 120 x + 312 x^2 - 383 x^3 + 204 x^4 - 40 x^6 + 12 x^7) h3[x_] := 3 - 20 x + 124 x^2 - 328 x^3 + 589 x^4 - 648 x^5 + 392 x^6 - 112 x^7 + 9 x^8 h4[x_] := 3 - 8 x + 18 x^2 + 112 x^3 - 395 x^4 + 684 x^5 - 644 x^6 + 296 x^7 - 51 x^8 h5[x_] := 3 - 8 x + 36 x^2 - 12 x^3 + 88 x^4 - 336 x^5 + 504 x^6 - 328 x^7 + 78 x^8 h6[x_] := 3 - 8 x + 36 x^2 - 3 x^4 + 56 x^5 - 196 x^6 + 200 x^7 - 66 x^8 h7[x_] := 3 - 8 x + 36 x^2 + 28 x^6 - 64 x^7 + 33 x^8 h8[x_] := 3 - 8 x + 36 x^2 + 8 x^7 - 9 x^8 h9[x_] := 3 - 8 x + 36 x^2 + x^8 h10[x_] := 3 - 8 x + 36 x^2 Factor[(1/(1-m)^2) B281[1-m, 1-x+m x] - (h0[x] (1-m) + h1[x] (1-m) m + h2[x] (1-m) m^2 + h3[x] (1-m) m^3 + h4[x] (1-m) m^4 + h5[x] (1-m) m^5 + h6[x] (1-m) m^6 + h7[x] (1-m) m^7 + h8[x] (1-m) m^8 + h9[x] (1-m) m^9 + h10[x] m^(10))] Plot[h0[s], {s, 0, 1}] Plot[h1[s], {s, 0, 1}] Plot[h2[s], {s, 0, 1}] Plot[h3[s], {s, 0, 1}] Plot[h4[s], {s, 0, 1}] Plot[h5[s], {s, 0, 1}] Plot[h6[s], {s, 0, 1}] Plot[h7[s], {s, 0, 1}] Plot[h8[s], {s, 0, 1}] Plot[h9[s], {s, 0, 1}] Plot[h10[s], {s, 0, 1}] (* End of File. *)