(* This file is for the following artice: Tetsuya ANDO, Extremal Cubic Inequalities of Three Variables. Copy and past suitable part, and test by Mathematica. *) (* Section 1 *) Hpq[x_,y_,z_,p_,q_] := 2 x^3 + 3(p-q)x^2 y - 6 p q x y^2 + q^2(3p+q)y^3 + 3(q-p)x^2z + 12 p q x y z + (p^3+3p^2q-6p q^2-2q^3) y^2 z - 6p q x z^2 + (-2p^3-6p^2q+3p q^2+q^3) y z^2 + p^2(p+3q)z^3 Gpq[x_,y_,z_,p_,q_] := z^3 + q^2x^2z + p^2y^2z - 2q x z^2 - 2p y z^2 + (1-p+q)(1-p-q)x^2y + (1+p-q)(1-p-q)x y^2 - (p^2+q^2-4p-4q+3)x y z a1[p_,q_] := p q-p+1 a2[p_,q_,r_] := p^2q r-p^2r+p q r-p q+2p r+p-r+1 c1[p_,q_,r_] := q^2 a1[p,q] a1[r,p] a2[p,q,r] (* c1[q,r,p] = r^2 a1[r,q] a1[p,q] a2[q,r,p] c1[r,p,q] = p^2 a1[r,p] a1[q,r] a2[r,p,q] *) c2[p_,q_,r_] := - a1[p,q](2p^3q^3r^3 - 2p^3q^2r^3 + 6p^2q^2r^3 - 2p q^3r^3 + 3p q^3r^2 - 6p q^2r^3 + 3p q^2r^2 + 2q^2r^3 - p q^3 - 3q^2r^2 + 3p q^2 - 3p q + q^2 + p - 1) c3[p_,q_,r_] := r a1[p,q](p^3q^3r^3 - p^3q^2r^3 + 3p^2q^2r^3 - p q^3r^3 - 3p q^2r^3 + 3p q^3r + q^2r^3 - 2p q^3 - 3p q^2r + 6p q^2 - 3q^2r - 6p q + 2q^2 + 2p - 2) c4[p_,q_,r_] := -c1[p,q,r]-c1[q,r,p]-c1[r,p,q]- c2[p,q,r]-c2[q,r,p]-c2[r,p,q]-c3[p,q,r]-c3[q,r,p]-c3[r,p,q] Fpqr[x_,y_,z_,p_,q_,r_] := c1[p,q,r]x^3 + c1[q,r,p]y^3+ c1[r,p,q]z^3 + c2[p,q,r]x^2y + c3[p,q,r]x y^2 + c2[q,r,p]y^2z + c3[q,r,p]y z^2 + c2[r,p,q]z^2x + c3[r,p,q]z x^2 + c4[p,q,r]x y z (*=========== Differential equations =========================*) f1[x_,y_,z_]:=x^3 f2[x_,y_,z_]:=x^2y f3[x_,y_,z_]:=x^2z f4[x_,y_,z_]:=x y^2 f5[x_,y_,z_]:=x y z f6[x_,y_,z_]:=x z^2 f7[x_,y_,z_]:=y^3 f8[x_,y_,z_]:=y^2z f9[x_,y_,z_]:=y z^2 f10[x_,y_,z_]:=z^3 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]} fx1[x_,y_,z_]:=3x^2 fx2[x_,y_,z_]:=2x y fx3[x_,y_,z_]:=2x z fx4[x_,y_,z_]:=y^2 fx5[x_,y_,z_]:=y z fx6[x_,y_,z_]:=z^2 fx7[x_,y_,z_]:=0 fx8[x_,y_,z_]:=0 fx9[x_,y_,z_]:=0 fx10[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]} fy1[x_,y_,z_]:=0 fy2[x_,y_,z_]:=x^2 fy3[x_,y_,z_]:=0 fy4[x_,y_,z_]:=2x y fy5[x_,y_,z_]:=x z fy6[x_,y_,z_]:=0 fy7[x_,y_,z_]:=3y^2 fy8[x_,y_,z_]:=2y z fy9[x_,y_,z_]:=z^2 fy10[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]} fz1[x_,y_,z_]:=0 fz2[x_,y_,z_]:=0 fz3[x_,y_,z_]:=x^2 fz4[x_,y_,z_]:=0 fz5[x_,y_,z_]:=x y fz6[x_,y_,z_]:=2x z fz7[x_,y_,z_]:=0 fz8[x_,y_,z_]:=y^2 fz9[x_,y_,z_]:=2y z fz10[x_,y_,z_]:=3z^2 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]} fxx1[x_,y_,z_]:=6x fxx2[x_,y_,z_]:=2y fxx3[x_,y_,z_]:=2z fxx4[x_,y_,z_]:=0 fxx5[x_,y_,z_]:=0 fxx6[x_,y_,z_]:=0 fxx7[x_,y_,z_]:=0 fxx8[x_,y_,z_]:=0 fxx9[x_,y_,z_]:=0 fxx10[x_,y_,z_]:=0 fxx[x_,y_,z_] := {fxx1[x,y,z],fxx2[x,y,z],fxx3[x,y,z],fxx4[x,y,z],fxx5[x,y,z],fxx6[x,y,z],fxx7[x,y,z],fxx8[x,y,z],fxx9[x,y,z],fxx10[x,y,z]} fyy1[x_,y_,z_]:=0 fyy2[x_,y_,z_]:=0 fyy3[x_,y_,z_]:=0 fyy4[x_,y_,z_]:=2x fyy5[x_,y_,z_]:=0 fyy6[x_,y_,z_]:=0 fyy7[x_,y_,z_]:=6y fyy8[x_,y_,z_]:=2z fyy9[x_,y_,z_]:=0 fyy10[x_,y_,z_]:=0 fyy[x_,y_,z_] := {fyy1[x,y,z],fyy2[x,y,z],fyy3[x,y,z],fyy4[x,y,z],fyy5[x,y,z],fyy6[x,y,z],fyy7[x,y,z],fyy8[x,y,z],fyy9[x,y,z],fyy10[x,y,z]} fzz1[x_,y_,z_]:=0 fzz2[x_,y_,z_]:=0 fzz3[x_,y_,z_]:=0 fzz4[x_,y_,z_]:=0 fzz5[x_,y_,z_]:=0 fzz6[x_,y_,z_]:=2x fzz7[x_,y_,z_]:=0 fzz8[x_,y_,z_]:=0 fzz9[x_,y_,z_]:=2y fzz10[x_,y_,z_]:=6z fzz[x_,y_,z_] := {fzz1[x,y,z],fzz2[x,y,z],fzz3[x,y,z],fzz4[x,y,z],fzz5[x,y,z],fzz6[x,y,z],fzz7[x,y,z],fzz8[x,y,z],fzz9[x,y,z],fzz10[x,y,z]} fxy1[x_,y_,z_]:=0 fxy2[x_,y_,z_]:=2x fxy3[x_,y_,z_]:=0 fxy4[x_,y_,z_]:=2y fxy5[x_,y_,z_]:=z fxy6[x_,y_,z_]:=0 fxy7[x_,y_,z_]:=0 fxy8[x_,y_,z_]:=0 fxy9[x_,y_,z_]:=0 fxy10[x_,y_,z_]:=0 fxy[x_,y_,z_] := {fxy1[x,y,z],fxy2[x,y,z],fxy3[x,y,z],fxy4[x,y,z],fxy5[x,y,z],fxy6[x,y,z],fxy7[x,y,z],fxy8[x,y,z],fxy9[x,y,z],fxy10[x,y,z]} fyx[x_,y_,z_] := fxy[x,y,z] fyz1[x_,y_,z_]:=0 fyz2[x_,y_,z_]:=0 fyz3[x_,y_,z_]:=0 fyz4[x_,y_,z_]:=0 fyz5[x_,y_,z_]:=x fyz6[x_,y_,z_]:=0 fyz7[x_,y_,z_]:=0 fyz8[x_,y_,z_]:=2y fyz9[x_,y_,z_]:=2z fyz10[x_,y_,z_]:=0 fyz[x_,y_,z_] := {fyz1[x,y,z],fyz2[x,y,z],fyz3[x,y,z],fyz4[x,y,z],fyz5[x,y,z],fyz6[x,y,z],fyz7[x,y,z],fyz8[x,y,z],fyz9[x,y,z],fyz10[x,y,z]} fzy[x_,y_,z_] := fyz[x,y,z] fzx1[x_,y_,z_]:=0 fzx2[x_,y_,z_]:=0 fzx3[x_,y_,z_]:=2x fzx4[x_,y_,z_]:=0 fzx5[x_,y_,z_]:=y fzx6[x_,y_,z_]:=2z fzx7[x_,y_,z_]:=0 fzx8[x_,y_,z_]:=0 fzx9[x_,y_,z_]:=0 fzx10[x_,y_,z_]:=0 fzx[x_,y_,z_] := {fzx1[x,y,z],fzx2[x,y,z],fzx3[x,y,z],fzx4[x,y,z],fzx5[x,y,z],fzx6[x,y,z],fzx7[x,y,z],fzx8[x,y,z],fzx9[x,y,z],fzx10[x,y,z]} fxz[x_,y_,z_] := fzx[x,y,z] (*=========================================================================*) (* Lemma 3.11 *) G[x_,y_,z_] := y(r x-y-z)^2 Factor[D[G[x,y,z],x]] (* = 2r y (r x-y-z) *) Factor[D[G[x,y,z],y]] (* = (r x-y-z)(r x-3y-z) *) Factor[D[G[x,y,z],z]] (* = -2y(r x-y-z) *) Factor[D[D[G[x,y,z],x],x]] (* = 2r^2y *) Factor[D[D[G[x,y,z],y],y]] (* = 2(-2r x+3y+2z) *) Factor[D[D[G[x,y,z],z],z]] (* = 2y *) H[x_,y_,z_] := y(y-z)^2 Factor[D[H[x,y,z],x]] (* = 0 *) Factor[D[H[x,y,z],y]] (* = (y-z)(3y-z) *) Factor[D[H[x,y,z],z]] (* = -2y(y-z) *) Factor[D[D[H[x,y,z],x],x]] (* = 0 *) Factor[D[D[H[x,y,z],y],y]] (* = 2(3y-2z) *) Factor[D[D[H[x,y,z],z],z]] (* = 2y *) (*=========================================================================*) (* Lemma 3.12 *) A312[p_,q_]:={f[0,1,1],fx[0,1,1],fy[0,1,1],fxy[0,1,1], fxx[0,1,1], f[p,0,1],fx[p,0,1],f[q,1,0],fy[q,1,0]} Factor[NullSpace[A312[p,q]]] Hpq[x_,y_,z_,p_,q_] := 2 x^3 + 3(p-q)x^2 y - 6 p q x y^2 + q^2(3p+q)y^3 + 3(q-p)x^2z + 12 p q x y z + (p^3+3p^2q-6p q^2-2q^3) y^2 z - 6p q x z^2 + (-2p^3-6p^2q+3p q^2+q^3) y z^2 + p^2(p+3q)z^3 Factor[Hpq[x,1,1+t x,p,q]] xH[t_,p_,q_] := -(((p + q)^3 t^2)/((-1 + p t)^2 (2 + p t + 3 q t))) Factor[1+t xH[t,p,q]] zH[t_,p_,q_] := -(((1 + q t)^2 (-2 + 3p t + q t))/((-1 + p t)^2 (2 + p t + 3q t))) Factor[Hpq[xH[t,p,q], 1, zH[t,p,q], p, q]] (* = 0 *) (*=========================================================================*) (* Theorem 3.13 *) (* (2) *) G[x_,y_,z_] := z(y-z)^2 Factor[D[G[x,y,z],x]] (* = 0 *) Factor[D[G[x,y,z],y]] (* = (x-y)(3y-z) *) Factor[D[G[x,y,z],z]] (* =-2y(y-z) *) Factor[D[D[G[x,y,z],x],x]] (* = 0 *) Factor[D[D[G[x,y,z],y],y]] (* = 2(3y-2z) *) Factor[D[D[G[x,y,z],z],z]] (* = 2y *) (* (3) *) A312[p_,q_]:={f[0,1,1],fx[0,1,1],fy[0,1,1],fxy[0,1,1], fxx[0,1,1], f[p,0,1],fx[p,0,1],f[0,1,0]} Factor[NullSpace[A312[p,q]]] w1[x_,y_,z_,p_] := (2 x^3 + 3 p x^2 y - 3 p x^2 z + p^3 y^2 z - 2 p^3 y z^2 + p^3 z^3) w2[x_,y_,z_,p_] := (x (x + p y - p z)^2) Factor[w1[x, y, z, p] - Hpq[x, y, z, p, 0]] (* = 0 *) (*=========================================================================*) (* Lemma 3.14 *) (* (1) *) A314[p_,q_] := {f[1,1,1], fx[1,1,1], fy[1,1,1], f[0,1,p], fy[0,1,p], f[1,0,q], fx[1,0,q], f[1,0,0], f[0,1,0]} Factor[NullSpace[A314[p, q]]] Factor[{0, (-1 + p - q) (-1 + p + q), q^2, -(1 + p - q) (-1 + p + q), -3 + 4 p - p^2 + 4 q - q^2, -2 q, 0, p^2, -2 p, 1}.f[x, y, z]] Gpq[x_,y_,z_,p_,q_] := z^3 + q^2x^2z + p^2y^2z - 2q x z^2 - 2p y z^2 + (1-p+q)(1-p-q)x^2y + (1+p-q)(1-p-q)x y^2 - (p^2+q^2-4p-4q+3)x y z (* Note that: *) e := {0,0,0,0,0, 0,0,0,0,1} Det[{e3,f[1,1,1],fx[1,1,1],fy[1,1,1], f[0,1,p],fy[0,1,p],f[1,0,q],fx[1,0,q],f[1,0,0],f[0,1,0]}] (*= p^3 q^3 *) (* $$f(P) = f(Q_x) = f(P_x) = f(P_y) = f_z(P_y) = f_{zz}(P_y) = 0,$$ *) (* (2) *) A314[p_] := {f[1,1,1], fx[1,1,1], fy[1,1,1], f[0,1,p], fy[0,1,p], f[0,1,0], f[1,0,0], fz[1,0,0], fzz[1,0,0]} Factor[NullSpace[A314[p]]] Factor[{0, (-1 + p)^2, 0, -(-1 + p) (1 + p), -(-3 + p) (-1 + p), 0, 0, p^2, -2 p, 1}.f[x, y, z] - Gpq[x, y, z, p, 0]] e := {0,0,0,0,0, 0,0,0,0,1} Det[{e,f[1,1,1], fx[1,1,1], fy[1,1,1], f[0,1,p], fy[0,1,p], f[0,1,0], f[1,0,0], fz[1,0,0], fzz[1,0,0]}] (* = 2p^3 *) (* (3) *) A314 := {f[1,1,1], fx[1,1,1], fy[1,1,1], f[1,0,0],fz[1,0,0],fzz[1,0,0], f[0,1,0], fz[0,1,0], fzz[0,1,0]} Factor[NullSpace[A314]] Factor[{0,1,0,1,-3,0,0,0,0,1}.f[x,y,z] - Gpq[x,y,z,0,0]] (* =0 *) (*--------------------------------------------------------------------------*) (* Theorem 3.15 *) Factor[Gpq[1+s, 1+s t,1,p,q]] Factor[Solve[(1 - 2 p + p^2 + t - p^2 t - q^2 t + s t - 2 p s t + p^2 s t - q^2 s t + t^2 - 2 q t^2 + q^2 t^2 + s t^2 - p^2 s t^2 - 2 q s t^2 + q^2 s t^2) == 0, s]] s[t_] := -((-1 + 2 p - p^2 - t + p^2 t + q^2 t - t^2 + 2 q t^2 - q^2 t^2)/((-1 + p + q) t (1 - p + q + t + p t - q t))) xG[t_] := (1-q+q t)^2/(t (1-p-q)(-(1+p-q) - t(1-p+q))) yG[t_] := (p+t-p t)^2/((1-p-q)(-(1+p-q) - t(1-p+q))) Factor[Gpq[xG[t], yG[t], 1, p, q]] (* = 0 *) (* Cf. *) Factor[Gpq[x,x,1,p,q]] (* = (x-1)^2(1-2(p+q-1)x) *) Factor[Gpq[x,y,z,p,1-p]] (* = ((p-1) x - p y - z)^2 z *) g[x_,y_,z_,p_,q_] := (1-p+q)x + (1+p-q)y - 2z Factor[Gpq[x,y,z,p,q] - ((1-p-q)y(x-z)g[x,y,z,p,q] + (q x + (1-q)y - z)^2 z)] (* = 0 *) Factor[Gpq[x,y,z,p,q] - ((1-p-q)x(y-z)g[x,y,z,p,q] + ((1-p)x + p y - z)^2 z)] (* = 0 *) Factor[Gpq[y,x,z,q,p] - Gpq[x,y,z,p,q]] (* = 0 *) (*=========================================================================*) (* Defintion 3.16. $f_{pqr}(x,y,z)$ *) a1[p_,q_] := p q-p+1 a2[p_,q_,r_] := p^2q r-p^2r+p q r-p q+2p r+p-r+1 c1[p_,q_,r_] := q^2 a1[p,q] a1[r,p] a2[p,q,r] (* c1[q,r,p] = r^2 a1[r,q] a1[p,q] a2[q,r,p] c1[r,p,q] = p^2 a1[r,p] a1[q,r] a2[r,p,q] *) c2[p_,q_,r_] := - a1[p,q](2p^3q^3r^3 - 2p^3q^2r^3 + 6p^2q^2r^3 - 2p q^3r^3 + 3p q^3r^2 - 6p q^2r^3 + 3p q^2r^2 + 2q^2r^3 - p q^3 - 3q^2r^2 + 3p q^2 - 3p q + q^2 + p - 1) c3[p_,q_,r_] := r a1[p,q](p^3q^3r^3 - p^3q^2r^3 + 3p^2q^2r^3 - p q^3r^3 - 3p q^2r^3 + 3p q^3r + q^2r^3 - 2p q^3 - 3p q^2r + 6p q^2 - 3q^2r - 6p q + 2q^2 + 2p - 2) c4[p_,q_,r_] := -c1[p,q,r]-c1[q,r,p]-c1[r,p,q]- c2[p,q,r]-c2[q,r,p]-c2[r,p,q]-c3[p,q,r]-c3[q,r,p]-c3[r,p,q] Fpqr[x_,y_,z_,p_,q_,r_] := c1[p,q,r]x^3 + c1[q,r,p]y^3+ c1[r,p,q]z^3 + c2[p,q,r]x^2y + c3[p,q,r]x y^2 + c2[q,r,p]y^2z + c3[q,r,p]y z^2 + c2[r,p,q]z^2x + c3[r,p,q]z x^2 + c4[p,q,r]x y z (*--------------------------------------------------------------------------*) (* Lemma 3.17. *) (* (1) *) A317[p_,q_,r_]:={f[1,1,1],fx[1,1,1],fy[1,1,1], f[0,p,1],fy[0,p,1], f[1,0,q],fz[1,0,q], f[r,1,0],fx[r,1,0]} Factor[NullSpace[A317[p,q,r]]] (* After obaining the solution, we put $f_{p,q}(x,y,y) to be: *) (* f0[x_,y_,z_,p_,q_,r_] := {The above solution}.f[x, y, z] *) (* Factor[Fpqr[x,y,z,p,q,r] - (p^2 a1[r,p] a1[q,r] (1 - q + r - p r + 2 q r + p q r - q r^2 + p q r^2)f0[x,y,z,p,q,r])] If not zero, multiply a certain constant! *) (* (2) *) A317[p_,q_]:={f[1,1,1],fx[1,1,1],fy[1,1,1], f[0,p,1],fy[0,p,1], f[1,0,q],fz[1,0,q], f[0,1,0],fx[0,1,0]} Factor[NullSpace[A317[p,q]]] (* (3) *) A317[p_]:={f[1,1,1],fx[1,1,1],fy[1,1,1], f[0,p,1],fy[0,p,1], f[1,0,0],fz[1,0,0], f[0,1,0],fx[0,1,0]} Factor[NullSpace[A317[p]]] (* (4) *) A317:={f[1,1,1],fx[1,1,1],fy[1,1,1], f[0,0,1],fy[0,0,1], f[1,0,0],fz[1,0,0], f[0,1,0],fx[0,1,0]} Factor[NullSpace[A317]] (* Some special cases. *) Factor[Fpqr[x,y,z,p,q,(1-1/q)]] (* = (1/q^2)(1-p+p q)^4x((q-1)y+z-q x)^2 *) Factor[Fpqr[x,y,z,p,q,1/(1-p)]] (* = (1/(p-1)^4)(1-p+p q)^4y((p-1)x+y-p z)^2 *) Factor[Fpqr[x,y,z,p,(1-1/p),r]] (* = (1/p^2)(1-r+p r)^4z((p-1)x+y-p z)^2 *) Factor[Fpqr[x,y,z,p,1/(1-r),r]] (* = (1/(r-1)^4)(1-r+p r)^4x((r-1)x+z-r z)^2 *) (*--------------------------------------------------------------------------*) (* Lemma 3.18. *) (* (1) *) Factor[a2[p,q,r] - (p r a1[p,q] + p a1[q,r]+ a1[r,p])] (* = 0 *) (* (2-2) *) b1[p_,q_] := -p^2q + p^2 - p q - 2p + 1 b2[p_,q_] := p^2q^2 - 2 p^2 q - 2 p q + p^2 - 2p + 1 r0[p_,q_] := (1 + p(1-q))/b1[p,q] r2[p_,q_] := (p(1-q)^2-1-q)/(q(p q+(p-1))) Factor[a2[p,q,r] - (-b1[p,q] r + a2[p,q,0])] Factor[a2[q,r,p] - (r q(p q+(p-1)) + (-p q^2 + 2p q - p + q + 1))] Factor[a2[r,p,q] - ((p-1)q r^2 - ((p-1)-(p+2)q)r + (1-q))] Factor[(r2[p,q] - r0[p,q]) - (-a1[p,q] b2[p,q])/(q (p q+(p-1)) b1[p,q])] Factor[a2[r0[p,q],p,q] - (2 a1[p,q] b2[p,q])/(((p-1)^2-p q(p+1))^2)] Factor[a2[r2[p,q],p,q] - 2a1[p,q] b2[p,q]/(((p-1)+p q)^2)] (*--------------------------------------------------------------------------*) (* Lemma 3.19. *) Factor[p^4 q^4 r^4 Fpqr[x,y,z,1/p,1/q,1/r] - Fpqr[x,z,y,p,r,q]] (*= 0 *) Factor[Fpqr[z,y,x,p,q,r] - p^4q^4r^4 Fpqr[x,y,z,1/r,1/q,1/p]] (* = 0 *) Factor[p q a1[1/p,1/q] - a1[q,p]] Factor[p^2 q r a2[1/p,1/q,1/r] - a2[p,r,q]] Factor[p^4q^4r^4 c1[1/p,1/q,1/r] - c1[p,r,q]] Factor[p^4q^4r^4 c2[1/p,1/q,1/r] - c3[q,p,r]] Factor[p^4q^4r^4 c3[1/p,1/q,1/r] - c2[q,p,r]] Factor[p^4q^4r^4 c4[1/p,1/q,1/r] - c4[p,r,q]] Factor[Fpqr[z,x,y,p,q,r] - Fpqr[x,y,z,q,r,p]] Factor[Fpqr[z,y,x,p,q,r] - Fpqr[x,z,y,r,p,q]] (*--------------------------------------------------------------------------*) (* Theorem 3.20. *) xpqr[t_,p_,q_,r_] := a1[q,r] (t+(p-1))^2 (r^2 a1[p,q] a2[q,r,p] t - ((p^2q^2r^2+1) a1[q,r] + 2q r a1[r,p] + 2p q r^2 a1[p,q])) ypqr[t_,p_,q_,r_] := a1[r,p] ((1-q)t+q)^2 (-((p^2q^2r^2+1)a1[r,p] + + 2p r a1[p,q] + 2p^2q r a1[q,r])t + a1[p,q] a2[p,q,r]) zpqr[t_,p_,q_,r_] := (r t-1)^2 a1[p,q] (a1[q,r] a2[q,r,p] t + q^2 a1[r,p] a2[p,q,r] ) Factor[Fpqr[xpqr[t,p,q,r],ypqr[t,p,q,r],zpqr[t,p,q,r],p,q,r]] (* = 0 *) t1 := ((p^2q^2r^2+1) a1[q,r] + 2q r a1[r,p] + 2p q r^2 a1[p,q]) / (r^2 a1[p,q] a2[q,r,p]) t2 := (a1[p,q] a2[p,q,r]) / ((p^2q^2r^2+1)a1[r,p] + 2p r a1[p,q] + 2p^2q r a1[q,r]) Factor[xpqr[t1,p,q,r]] Factor[ypqr[t2,p,q,r]] Factor[(t1 - t2) - (a2[r,p,q](a1[r,p] + p r a1[p,q] + p^2q r a1[q,r])^2) / ( r^2 a1[p,q] a2[q,r,p] (a1[r,p] + 2p r a1[p,q] + 2p^2q r a1[q,r] + p^2q^2r^2a1[r,p]))] (*=========================================================================*) (* Theorem 3.22 *) A322[p_,q_]:={f[1,1,1],fx[1,1,1],fy[1,1,1], f[0,p,1],fy[0,p,1], f[q,0,1],fz[q,0,1]} Factor[NullSpace[A322[p,q]]] Factor[{2/q^3, (-4 p^3 + 3 p^3 q + 2 q^3 - 3 p q^3 + p^3 q^3)/( p^3 q^3), -(3/q^2), (2 p^3 - 3 p^3 q - 4 q^3 + 3 p q^3 + p^3 q^3)/( p^3 q^3), -((3 (-p^2 - q^2 + p^2 q^2))/(p^2 q^2)), 0, 2/ p^3, -(3/p^2), 0, 1}.f[x, y, z]] Factor[{0, (-1 + p)^2/p^2, 0, (2 (-1 + p))/p^2, -((2 (-1 + p))/p), 0, 1/p^2, -(2/p), 1, 0}.f[x, y, z]] Factor[{1/q^2, (2 (-1 + q))/q^2, -(2/q), (-1 + q)^2/ q^2, -((2 (-1 + q))/q), 1, 0, 0, 0, 0}.f[x, y, z]] tm[x_,y_,z_,p_,q_] := (2 p^3 x^3 - 4 p^3 x^2 y + 3 p^3 q x^2 y + 2 q^3 x^2 y - 3 p q^3 x^2 y + p^3 q^3 x^2 y + 2 p^3 x y^2 - 3 p^3 q x y^2 - 4 q^3 x y^2 + 3 p q^3 x y^2 + p^3 q^3 x y^2 + 2 q^3 y^3 - 3 p^3 q x^2 z + 3 p^3 q x y z + 3 p q^3 x y z - 3 p^3 q^3 x y z - 3 p q^3 y^2 z + p^3 q^3 z^3) y((p-1)x + y - p z)^2 x(x + (q-1)y - q z)^2 (* (0) *) A322z[p_,q_]:={f[1,1,1],fx[1,1,1],fy[1,1,1], f[0,p,1],fy[0,p,1], f[q,0,1],fz[q,0,1], f[0,0,1]} Factor[NullSpace[A322z[p,q]]] Factor[{0, (-1 + p)^2/p^2, 0, (2 (-1 + p))/p^2, -((2 (-1 + p))/p), 0, 1/p^2, -(2/p), 1, 0}.f[x, y, z]] Factor[{q^2, -2 (-1 + q) q, -2 q, (-1 + q)^2, 2 (-1 + q), 1, 0, 0, 0, 0}.f[x, y, z]] y((p-1)x + y - p z)^2 x (x + (q-1)y - qz)^2 (* (2) *) A322c[p_,q_]:={f[1,1,1],fx[1,1,1],fy[1,1,1], f[0,p,1],fy[0,p,1], f[q,0,1],fz[q,0,1], f[1,0,0],fy[1,0,0]} Factor[NullSpace[A322c[p,q]]] Factor[{0, 0, 1/q^2, ((-p - q + p q) (-p - p^2 + q - 2 p q + p^2 q))/((-1 + p) p^2 q^2), -((-p^2 + 3 p^3 + 4 p^2 q - 4 p^3 q - q^2 + 3 p q^2 - 3 p^2 q^2 + p^3 q^2)/((-1 + p) p^2 q^2)), -(2/ q), -(((-p - q + p q) (p - q + p q))/((-1 + p)^2 p^2 q^2)), (-2 p^3 + q^2 - 3 p^2 q^2 + 2 p^3 q^2)/((-1 + p)^2 p^2 q^2), -((-p^3 + 2 q^2 - 3 p q^2 + p^3 q^2)/((-1 + p)^2 p q^2)), 1}.f[x, y, z]] wq := (-p^2 x y^2 + p^4 x y^2 - 2 p^2 q x y^2 + 4 p^3 q x y^2 - 2 p^4 q x y^2 + q^2 x y^2 - 4 p q^2 x y^2 + 6 p^2 q^2 x y^2 - 4 p^3 q^2 x y^2 + p^4 q^2 x y^2 + p^2 y^3 - q^2 y^3 + 2 p q^2 y^3 - p^2 q^2 y^3 + p^2 x^2 z - 2 p^3 x^2 z + p^4 x^2 z - p^2 x y z + 4 p^3 x y z - 3 p^4 x y z + 4 p^2 q x y z - 8 p^3 q x y z + 4 p^4 q x y z - q^2 x y z + 4 p q^2 x y z - 6 p^2 q^2 x y z + 4 p^3 q^2 x y z - p^4 q^2 x y z - 2 p^3 y^2 z + q^2 y^2 z - 3 p^2 q^2 y^2 z + 2 p^3 q^2 y^2 z - 2 p^2 q x z^2 + 4 p^3 q x z^2 - 2 p^4 q x z^2 + p^4 y z^2 - 2 p q^2 y z^2 + 3 p^2 q^2 y z^2 - p^4 q^2 y z^2 + p^2 q^2 z^3 - 2 p^3 q^2 z^3 + p^4 q^2 z^3) Factor[wq - p^4 fpqr[x, z, y, 1/p, 0, q]] (* = 0 *) (*=========================================================================*) (* Theorem 3.23 *) A323[p_]:={f[1,1,1],fx[1,1,1],fy[1,1,1],f[0,p,1],fy[0,p,1], f[1,0,0], f[0,1,0]} Factor[NullSpace[A323[p]]] Factor[{0, (2 (-1 + p) (-1 + 2 p))/ p^2, -(((-1 + p) (-1 + 3 p))/p^2), -((2 (-1 + p)^2)/p^2), 0, 0, 0, 1/p^2, -(2/p), 1}.f[x, y, z]] Factor[{0, 2, -2, -1, 0, 1, 0, 0, 0, 0}.f[x, y, z]] Factor[{0, 1, -1, -1, 1, 0, 0, 0, 0, 0}.f[x, y, z]] (2 x^2 y - 6 p x^2 y + 4 p^2 x^2 y - 2 x y^2 + 4 p x y^2 - 2 p^2 x y^2 - x^2 z + 4 p x^2 z - 3 p^2 x^2 z + y^2 z - 2 p y z^2 + p^2 z^3) x (2 x - y - z) (y - z) x (x - y) (y - z) (* (i-1) *) A323a[p_]:={f[1,1,1],fx[1,1,1],fy[1,1,1],f[0,p,1],fy[0,p,1], f[0,1,0], f[1,0,0], fz[1,0,0], fzz[1,0,0]} Factor[NullSpace[A323a[p]]] Factor[{0, (-1 + p)^2/p^2, 0, ((-1 + p) (1 + p))/ p^2, -(((-1 + p) (-1 + 3 p))/p^2), 0, 0, 1/p^2, -(2/p), 1}.f[x,y,z]] Factor[(1/p^2) (x^2 y - 2 p x^2 y + p^2 x^2 y - x y^2 + p^2 x y^2 - x y z + 4 p x y z - 3 p^2 x y z + y^2 z - 2 p y z^2 + p^2 z^3) - Gpq[x, y, z, 1/p, 0]] (* = 0 *) (* (i-2) *) A323b[p_]:={f[1,1,1],fx[1,1,1],fy[1,1,1],f[0,p,1],fy[0,p,1], f[0,1,0], fx[0,1,0], f[1,0,0], fz[1,0,0]} Factor[NullSpace[A323b[p]]] Factor[{0, (-1 + p)^2/p^2, 0, 0, -(((-3 + p) (-1 + p))/p^2), -(((-1 + p) (1 + p))/p^2), 0, 1/ p^2, -(2/p), 1}.f[x, y, z]] Factor[(x^2 y - 2 p x^2 y + p^2 x^2 y - 3 x y z + 4 p x y z - p^2 x y z + y^2 z + x z^2 - p^2 x z^2 - 2 p y z^2 + p^2 z^3) - fpqr[x, y, z, p, 0, 0]] (* = 0 *) (* (i-3) *) A323c[p_]:={f[1,1,1],fx[1,1,1],fy[1,1,1],f[0,p,1],fy[0,p,1], f[0,1,0], fx[0,1,0], fxx[0,1,0], f[1,0,0]} Factor[NullSpace[A323c[p]]] Factor[{0, 0, (-1 + p)^2/p^2, 0, (2 (-1 + p))/p^2, -((2 (-1 + p))/p), 0, 1/p^2, -(2/p), 1}.f[x, y, z]] z((p-1)x + y - p z)^2 (*=========================================================================*) (* Theorem 3.24 *) A324a:={f[1,1,1],fx[1,1,1],fy[1,1,1],f[1,0,0],fy[1,0,0], f[0,1,0],fz[0,1,0], f[0,0,1],fx[0,0,1]} Factor[NullSpace[A324a]] x y^2 + x^2 z - 3 x y z + y z^2 G[x_,y_,z_] := y(x-z)^2 Factor[D[G[x,y,z],x]] (* = 2y(x-z) *) Factor[D[G[x,y,z],y]] (* = (x-z)^2 *) Factor[D[G[x,y,z],z]] (* = -2y(x-z) *) Factor[D[D[G[x,y,z],x],x]] (* = 2y *) Factor[D[D[G[x,y,z],y],y]] (* = 0 *) Factor[D[D[G[x,y,z],z],z]] (* = 2y *) (*=========================================================================*) (* Theorem 4.2 *) f42[x_,y_,z_] := 591900050 x^4 + 437205100 x^3 y - 766414561 x^2 y^2 + 217365672 x y^3 - 1650610670 x^3 z - 102695021 x^2 y z + 248518503 x y^2 z + 549666 y^3 z + 1531736792 x^2 z^2 + 118221267 x y z^2 + 101630538 y^2 z^2 - 636743352 x z^3 - 273946320 y z^3 + 183282336 z^4 (*------------------*) f1[x_,y_,z_]:=x^4 f2[x_,y_,z_]:=x^3y f3[x_,y_,z_]:=x^3z f4[x_,y_,z_]:=x^2y^2 f5[x_,y_,z_]:=x^2y z f6[x_,y_,z_]:=x^2z^2 f7[x_,y_,z_]:=x y^3 f8[x_,y_,z_]:=x y^2z f9[x_,y_,z_]:=x y z^2 f10[x_,y_,z_]:=x z^3 f11[x_,y_,z_]:=y^4 f12[x_,y_,z_]:=y^3z f13[x_,y_,z_]:=y^2z^2 f14[x_,y_,z_]:=y z^3 f15[x_,y_,z_]:=z^4 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]} fx1[x_,y_,z_]:=4x^3 fx2[x_,y_,z_]:=3x^2y fx3[x_,y_,z_]:=3x^2z fx4[x_,y_,z_]:=2x y^2 fx5[x_,y_,z_]:=2x y z fx6[x_,y_,z_]:=2x z^2 fx7[x_,y_,z_]:=y^3 fx8[x_,y_,z_]:=y^2z fx9[x_,y_,z_]:=y z^2 fx10[x_,y_,z_]:=z^3 fx11[x_,y_,z_]:=0 fx12[x_,y_,z_]:=0 fx13[x_,y_,z_]:=0 fx14[x_,y_,z_]:=0 fx15[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]} fy1[x_,y_,z_]:=0 fy2[x_,y_,z_]:=x^3 fy3[x_,y_,z_]:=0 fy4[x_,y_,z_]:=2x^2y fy5[x_,y_,z_]:=x^2 z fy6[x_,y_,z_]:=0 fy7[x_,y_,z_]:=3x y^2 fy8[x_,y_,z_]:=2x y z fy9[x_,y_,z_]:=x z^2 fy10[x_,y_,z_]:=0 fy11[x_,y_,z_]:=4y^3 fy12[x_,y_,z_]:=3y^2z fy13[x_,y_,z_]:=2y z^2 fy14[x_,y_,z_]:=z^3 fy15[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]} fz1[x_,y_,z_]:=0 fz2[x_,y_,z_]:=0 fz3[x_,y_,z_]:=x^3 fz4[x_,y_,z_]:=0 fz5[x_,y_,z_]:=x^2y fz6[x_,y_,z_]:=2x^2z fz7[x_,y_,z_]:=0 fz8[x_,y_,z_]:=x y^2 fz9[x_,y_,z_]:=2x y z fz10[x_,y_,z_]:=3x z^2 fz11[x_,y_,z_]:=0 fz12[x_,y_,z_]:=y^3 fz13[x_,y_,z_]:=2y^2z fz14[x_,y_,z_]:=3y z^2 fz15[x_,y_,z_]:=4z^3 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]} A42a := {f[1,1,1],fx[1,1,1],fy[1,1,1], f[2,3,1],fx[2,3,1],fy[2,3,1], f[1,2,3],fy[1,2,3],fz[1,2,3], f[0,4,3],fz[0,4,3], f[6,0,5],fx[6,0,5], f[0,1,0]} Factor[NullSpace[A42a]] (*------------------*) f1[x_,y_,z_] := x^8 f2[x_,y_,z_] := x^7y f3[x_,y_,z_] := x^7z f4[x_,y_,z_] := x^6y^2 f5[x_,y_,z_] := x^6y z f6[x_,y_,z_] := x^6z^2 f7[x_,y_,z_] := x^5y^3 f8[x_,y_,z_] := x^5y^2z f9[x_,y_,z_] := x^5y z^2 f10[x_,y_,z_] := x^5z^3 f11[x_,y_,z_] := x^4y^4 f12[x_,y_,z_] := x^4y^3z f13[x_,y_,z_] := x^4y^2z^2 f14[x_,y_,z_] := x^4y z^3 f15[x_,y_,z_] := x^4z^4 f16[x_,y_,z_] := x^3y^5 f17[x_,y_,z_] := x^3y^4z f18[x_,y_,z_] := x^3y^3z^2 f19[x_,y_,z_] := x^3y^2z^3 f20[x_,y_,z_] := x^3y z^4 f21[x_,y_,z_] := x^3z^5 f22[x_,y_,z_] := x^2y^6 f23[x_,y_,z_] := x^2y^5z f24[x_,y_,z_] := x^2y^4z^2 f25[x_,y_,z_] := x^2y^3z^3 f26[x_,y_,z_] := x^2y^2z^4 f27[x_,y_,z_] := x^2y z^5 f28[x_,y_,z_] := x^2z^6 f29[x_,y_,z_] := x y^7 f30[x_,y_,z_] := x y^6z f31[x_,y_,z_] := x y^5z^2 f32[x_,y_,z_] := x y^4z^3 f33[x_,y_,z_] := x y^3z^4 f34[x_,y_,z_] := x y^2z^5 f35[x_,y_,z_] := x y z^6 f36[x_,y_,z_] := x z^7 f37[x_,y_,z_] := y^8 f38[x_,y_,z_] := y^7z f39[x_,y_,z_] := y^6z^2 f40[x_,y_,z_] := y^5z^3 f41[x_,y_,z_] := y^4z^4 f42[x_,y_,z_] := y^3z^5 f43[x_,y_,z_] := y^2z^6 f44[x_,y_,z_] := y z^7 f45[x_,y_,z_] := z^8 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]} fx1[x_,y_,z_] := 8 x^7 fx2[x_,y_,z_] := 7 x^6y fx3[x_,y_,z_] := 7 x^6z fx4[x_,y_,z_] := 6 x^5y^2 fx5[x_,y_,z_] := 6 x^5y z fx6[x_,y_,z_] := 6 x^5z^2 fx7[x_,y_,z_] := 5 x^4y^3 fx8[x_,y_,z_] := 5 x^4y^2z fx9[x_,y_,z_] := 5 x^4y z^2 fx10[x_,y_,z_] := 5 x^4z^3 fx11[x_,y_,z_] := 4 x^3y^4 fx12[x_,y_,z_] := 4 x^3y^3z fx13[x_,y_,z_] := 4 x^3y^2z^2 fx14[x_,y_,z_] := 4 x^3y z^3 fx15[x_,y_,z_] := 4 x^3z^4 fx16[x_,y_,z_] := 3 x^2y^5 fx17[x_,y_,z_] := 3 x^2y^4z fx18[x_,y_,z_] := 3 x^2y^3z^2 fx19[x_,y_,z_] := 3 x^2y^2z^3 fx20[x_,y_,z_] := 3 x^2y z^4 fx21[x_,y_,z_] := 3 x^2z^5 fx22[x_,y_,z_] := 2 x y^6 fx23[x_,y_,z_] := 2 x y^5z fx24[x_,y_,z_] := 2 x y^4z^2 fx25[x_,y_,z_] := 2 x y^3z^3 fx26[x_,y_,z_] := 2 x y^2z^4 fx27[x_,y_,z_] := 2 x y z^5 fx28[x_,y_,z_] := 2 x z^6 fx29[x_,y_,z_] := y^7 fx30[x_,y_,z_] := y^6z fx31[x_,y_,z_] := y^5z^2 fx32[x_,y_,z_] := y^4z^3 fx33[x_,y_,z_] := y^3z^4 fx34[x_,y_,z_] := y^2z^5 fx35[x_,y_,z_] := y z^6 fx36[x_,y_,z_] := z^7 fx37[x_,y_,z_] := 0 fx38[x_,y_,z_] := 0 fx39[x_,y_,z_] := 0 fx40[x_,y_,z_] := 0 fx41[x_,y_,z_] := 0 fx42[x_,y_,z_] := 0 fx43[x_,y_,z_] := 0 fx44[x_,y_,z_] := 0 fx45[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]} fy1[x_,y_,z_] := 0 fy2[x_,y_,z_] := x^7 fy3[x_,y_,z_] := 0 fy4[x_,y_,z_] := 2 x^6y fy5[x_,y_,z_] := x^6z fy6[x_,y_,z_] := 0 fy7[x_,y_,z_] := 3 x^5y^2 fy8[x_,y_,z_] := 2 x^5y z fy9[x_,y_,z_] := x^5z^2 fy10[x_,y_,z_] := 0 fy11[x_,y_,z_] := 4 x^4y^3 fy12[x_,y_,z_] := 3 x^4y^2z fy13[x_,y_,z_] := 2 x^4y z^2 fy14[x_,y_,z_] := x^4z^3 fy15[x_,y_,z_] := 0 fy16[x_,y_,z_] := 5 x^3y^4 fy17[x_,y_,z_] := 4 x^3y^3z fy18[x_,y_,z_] := 3 x^3y^2z^2 fy19[x_,y_,z_] := 2 x^3y z^3 fy20[x_,y_,z_] := x^3z^4 fy21[x_,y_,z_] := 0 fy22[x_,y_,z_] := 6 x^2y^5 fy23[x_,y_,z_] := 5 x^2y^4z fy24[x_,y_,z_] := 4 x^2y^3z^2 fy25[x_,y_,z_] := 3 x^2y^2z^3 fy26[x_,y_,z_] := 2 x^2y z^4 fy27[x_,y_,z_] := x^2z^5 fy28[x_,y_,z_] := 0 fy29[x_,y_,z_] := 7 x y^6 fy30[x_,y_,z_] := 6 x y^5z fy31[x_,y_,z_] := 5 x y^4z^2 fy32[x_,y_,z_] := 4 x y^3z^3 fy33[x_,y_,z_] := 3 x y^2z^4 fy34[x_,y_,z_] := 2 x y z^5 fy35[x_,y_,z_] := x z^6 fy36[x_,y_,z_] := 0 fy37[x_,y_,z_] := 8 y^7 fy38[x_,y_,z_] := 7 y^6z fy39[x_,y_,z_] := 6 y^5z^2 fy40[x_,y_,z_] := 5 y^4z^3 fy41[x_,y_,z_] := 4 y^3z^4 fy42[x_,y_,z_] := 3 y^2z^5 fy43[x_,y_,z_] := 2 y z^6 fy44[x_,y_,z_] := z^7 fy45[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]} fz1[x_,y_,z_] := 0 fz2[x_,y_,z_] := 0 fz3[x_,y_,z_] := x^7 fz4[x_,y_,z_] := 0 fz5[x_,y_,z_] := x^6y fz6[x_,y_,z_] := 2 x^6z fz7[x_,y_,z_] := 0 fz8[x_,y_,z_] := x^5y^2 fz9[x_,y_,z_] := 2 x^5y z fz10[x_,y_,z_] := 3 x^5z^2 fz11[x_,y_,z_] := 0 fz12[x_,y_,z_] := x^4y^3 fz13[x_,y_,z_] := 2 x^4y^2z fz14[x_,y_,z_] := 3 x^4y z^2 fz15[x_,y_,z_] := 4 x^4z^3 fz16[x_,y_,z_] := 0 fz17[x_,y_,z_] := x^3y^4 fz18[x_,y_,z_] := 2 x^3y^3z fz19[x_,y_,z_] := 3 x^3y^2z^2 fz20[x_,y_,z_] := 4 x^3y z^3 fz21[x_,y_,z_] := 5 x^3z^4 fz22[x_,y_,z_] := 0 fz23[x_,y_,z_] := x^2y^5 fz24[x_,y_,z_] := 2 x^2y^4z fz25[x_,y_,z_] := 3 x^2y^3z^2 fz26[x_,y_,z_] := 4 x^2y^2z^3 fz27[x_,y_,z_] := 5 x^2y z^4 fz28[x_,y_,z_] := 6 x^2z^5 fz29[x_,y_,z_] := 0 fz30[x_,y_,z_] := x y^6 fz31[x_,y_,z_] := 2 x y^5z fz32[x_,y_,z_] := 3 x y^4z^2 fz33[x_,y_,z_] := 4 x y^3z^3 fz34[x_,y_,z_] := 5 x y^2z^4 fz35[x_,y_,z_] := 6 x y z^5 fz36[x_,y_,z_] := 7 x z^6 fz37[x_,y_,z_] := 0 fz38[x_,y_,z_] := y^7 fz39[x_,y_,z_] := 2 y^6z fz40[x_,y_,z_] := 3 y^5z^2 fz41[x_,y_,z_] := 4 y^4z^3 fz42[x_,y_,z_] := 5 y^3z^4 fz43[x_,y_,z_] := 6 y^2z^5 fz44[x_,y_,z_] := 7 y z^6 fz45[x_,y_,z_] := 8 z^7 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]} A42:={f[1,1,1],fx[1,1,1],fy[1,1,1], f[1,1,-1],fx[1,1,-1],fy[1,1,-1], f[1,-1,1],fx[1,-1,1],fy[1,-1,1], f[-1,1,1],fx[-1,1,1],fy[-1,1,1], f[Sqrt[2],Sqrt[3],1],fx[Sqrt[2],Sqrt[3],1],fy[Sqrt[2],Sqrt[3],1], f[Sqrt[2],Sqrt[3],-1],fx[Sqrt[2],Sqrt[3],-1],fy[Sqrt[2],Sqrt[3],-1], f[Sqrt[2],-Sqrt[3],1],fx[Sqrt[2],-Sqrt[3],1],fy[Sqrt[2],-Sqrt[3],1], f[-Sqrt[2],Sqrt[3],1],fx[-Sqrt[2],Sqrt[3],1],fy[-Sqrt[2],Sqrt[3],1], f[1,Sqrt[2],Sqrt[3]],fx[1,Sqrt[2],Sqrt[3]],fy[1,Sqrt[2],Sqrt[3]], f[1,Sqrt[2],-Sqrt[3]],fx[1,Sqrt[2],-Sqrt[3]],fy[1,Sqrt[2],-Sqrt[3]], f[1,-Sqrt[2],Sqrt[3]],fx[1,-Sqrt[2],Sqrt[3]],fy[1,-Sqrt[2],Sqrt[3]], f[-1,Sqrt[2],Sqrt[3]],fx[-1,Sqrt[2],Sqrt[3]],fy[-1,Sqrt[2],Sqrt[3]], f[0,2,Sqrt[3]],fx[0,2,Sqrt[3]],fy[0,2,Sqrt[3]], f[0,2,-Sqrt[3]],fx[0,2,-Sqrt[3]],fy[0,2,-Sqrt[3]], f[Sqrt[6],0,Sqrt[5]],fx[Sqrt[6],0,Sqrt[5]],fy[Sqrt[6],0,Sqrt[5]], f[Sqrt[6],0,-Sqrt[5]],fx[Sqrt[6],0,-Sqrt[5]],fy[Sqrt[6],0,-Sqrt[5]], f[0,1,0],fx[0,1,0],fy[0,1,0]} Factor[NullSpace[A42]] (*===========================================================================*) (* Theorem 4.3 *) f43[x_,y_,z_] := (837 x^4 y - 645 x^3 y^2 - 645 x^2 y^3 + 837 x y^4 + 1755 x^4 z - 17181 x^3 y z + 23876 x^2 y^2 z - 17181 x y^3 z + 1755 y^4 z - 3486 x^3 z^2 + 19594 x^2 y z^2 + 19594 x y^2 z^2 - 3486 y^3 z^2 + 3287 x^2 z^3 - 11030 x y z^3 + 3287 y^2 z^3 - 1692 x z^4 - 1692 y z^4 + 648 z^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]} A43a:= {f[4,1,1],fx[4,1,1],fy[4,1,1], f[1,4,1],fx[1,4,1],fy[1,4,1], f[1,1,4],fx[1,1,4],fy[1,1,4], f[1/9,1,1], fx[1/9,1,1], fy[1/9,1,1], f[1,1/9,1], fx[1,1/9,1], fy[1,1/9,1], f[1,1,1/9], fx[1,1,1/9], fy[1,1,1/9], f[1,0,0], f[0,1,0]} Factor[NullSpace[A43a]] (*------------------*) 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_] := 10 x^9 fx2[x_,y_,z_] := 9 x^8y fx3[x_,y_,z_] := 9 x^8z fx4[x_,y_,z_] := 8 x^7y^2 fx5[x_,y_,z_] := 8 x^7y z fx6[x_,y_,z_] := 8 x^7z^2 fx7[x_,y_,z_] := 7 x^6y^3 fx8[x_,y_,z_] := 7 x^6y^2z fx9[x_,y_,z_] := 7 x^6y z^2 fx10[x_,y_,z_] := 7 x^6z^3 fx11[x_,y_,z_] := 6 x^5y^4 fx12[x_,y_,z_] := 6 x^5y^3z fx13[x_,y_,z_] := 6 x^5y^2z^2 fx14[x_,y_,z_] := 6 x^5y z^3 fx15[x_,y_,z_] := 6 x^5z^4 fx16[x_,y_,z_] := 5 x^4y^5 fx17[x_,y_,z_] := 5 x^4y^4z fx18[x_,y_,z_] := 5 x^4y^3z^2 fx19[x_,y_,z_] := 5 x^4y^2z^3 fx20[x_,y_,z_] := 5 x^4y z^4 fx21[x_,y_,z_] := 5 x^4z^5 fx22[x_,y_,z_] := 4 x^3y^6 fx23[x_,y_,z_] := 4 x^3y^5z fx24[x_,y_,z_] := 4 x^3y^4z^2 fx25[x_,y_,z_] := 4 x^3y^3z^3 fx26[x_,y_,z_] := 4 x^3y^2z^4 fx27[x_,y_,z_] := 4 x^3y z^5 fx28[x_,y_,z_] := 4 x^3z^6 fx29[x_,y_,z_] := 3 x^2y^7 fx30[x_,y_,z_] := 3 x^2y^6z fx31[x_,y_,z_] := 3 x^2y^5z^2 fx32[x_,y_,z_] := 3 x^2y^4z^3 fx33[x_,y_,z_] := 3 x^2y^3z^4 fx34[x_,y_,z_] := 3 x^2y^2z^5 fx35[x_,y_,z_] := 3 x^2y z^6 fx36[x_,y_,z_] := 3 x^2z^7 fx37[x_,y_,z_] := 2 x y^8 fx38[x_,y_,z_] := 2 x y^7z fx39[x_,y_,z_] := 2 x y^6z^2 fx40[x_,y_,z_] := 2 x y^5z^3 fx41[x_,y_,z_] := 2 x y^4z^4 fx42[x_,y_,z_] := 2 x y^3z^5 fx43[x_,y_,z_] := 2 x y^2z^6 fx44[x_,y_,z_] := 2 x y z^7 fx45[x_,y_,z_] := 2 x 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_] := 2 x^8y fy5[x_,y_,z_] := x^8z fy6[x_,y_,z_] := 0 fy7[x_,y_,z_] := 3 x^7y^2 fy8[x_,y_,z_] := 2 x^7y z fy9[x_,y_,z_] := x^7z^2 fy10[x_,y_,z_] := 0 fy11[x_,y_,z_] := 4 x^6y^3 fy12[x_,y_,z_] := 3 x^6y^2z fy13[x_,y_,z_] := 2 x^6y z^2 fy14[x_,y_,z_] := x^6z^3 fy15[x_,y_,z_] := 0 fy16[x_,y_,z_] := 5 x^5y^4 fy17[x_,y_,z_] := 4 x^5y^3z fy18[x_,y_,z_] := 3 x^5y^2z^2 fy19[x_,y_,z_] := 2 x^5y z^3 fy20[x_,y_,z_] := x^5z^4 fy21[x_,y_,z_] := 0 fy22[x_,y_,z_] := 6 x^4y^5 fy23[x_,y_,z_] := 5 x^4y^4z fy24[x_,y_,z_] := 4 x^4y^3z^2 fy25[x_,y_,z_] := 3 x^4y^2z^3 fy26[x_,y_,z_] := 2 x^4y z^4 fy27[x_,y_,z_] := x^4z^5 fy28[x_,y_,z_] := 0 fy29[x_,y_,z_] := 7 x^3y^6 fy30[x_,y_,z_] := 6 x^3y^5z fy31[x_,y_,z_] := 5 x^3y^4z^2 fy32[x_,y_,z_] := 4 x^3y^3z^3 fy33[x_,y_,z_] := 3 x^3y^2z^4 fy34[x_,y_,z_] := 2 x^3y z^5 fy35[x_,y_,z_] := x^3z^6 fy36[x_,y_,z_] := 0 fy37[x_,y_,z_] := 8 x^2y^7 fy38[x_,y_,z_] := 7 x^2y^6z fy39[x_,y_,z_] := 6 x^2y^5z^2 fy40[x_,y_,z_] := 5 x^2y^4z^3 fy41[x_,y_,z_] := 4 x^2y^3z^4 fy42[x_,y_,z_] := 3 x^2y^2z^5 fy43[x_,y_,z_] := 2 x^2y z^6 fy44[x_,y_,z_] := x^2z^7 fy45[x_,y_,z_] := 0 fy46[x_,y_,z_] := 9 x y^8 fy47[x_,y_,z_] := 8 x y^7z fy48[x_,y_,z_] := 7 x y^6z^2 fy49[x_,y_,z_] := 6 x y^5z^3 fy50[x_,y_,z_] := 5 x y^4z^4 fy51[x_,y_,z_] := 4 x y^3z^5 fy52[x_,y_,z_] := 3 x y^2z^6 fy53[x_,y_,z_] := 2 x y z^7 fy54[x_,y_,z_] := x z^8 fy55[x_,y_,z_] := 0 fy56[x_,y_,z_] := 10 y^9 fy57[x_,y_,z_] := 9 y^8z fy58[x_,y_,z_] := 8 y^7z^2 fy59[x_,y_,z_] := 7 y^6z^3 fy60[x_,y_,z_] := 6 y^5z^4 fy61[x_,y_,z_] := 5 y^4z^5 fy62[x_,y_,z_] := 4 y^3z^6 fy63[x_,y_,z_] := 3 y^2z^7 fy64[x_,y_,z_] := 2 y 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_] := 2 x^8z fz7[x_,y_,z_] := 0 fz8[x_,y_,z_] := x^7y^2 fz9[x_,y_,z_] := 2 x^7y z fz10[x_,y_,z_] := 3 x^7z^2 fz11[x_,y_,z_] := 0 fz12[x_,y_,z_] := x^6y^3 fz13[x_,y_,z_] := 2 x^6y^2z fz14[x_,y_,z_] := 3 x^6y z^2 fz15[x_,y_,z_] := 4 x^6z^3 fz16[x_,y_,z_] := 0 fz17[x_,y_,z_] := x^5y^4 fz18[x_,y_,z_] := 2 x^5y^3z fz19[x_,y_,z_] := 3 x^5y^2z^2 fz20[x_,y_,z_] := 4 x^5y z^3 fz21[x_,y_,z_] := 5 x^5z^4 fz22[x_,y_,z_] := 0 fz23[x_,y_,z_] := x^4y^5 fz24[x_,y_,z_] := 2 x^4y^4z fz25[x_,y_,z_] := 3 x^4y^3z^2 fz26[x_,y_,z_] := 4 x^4y^2z^3 fz27[x_,y_,z_] := 5 x^4y z^4 fz28[x_,y_,z_] := 6 x^4z^5 fz29[x_,y_,z_] := 0 fz30[x_,y_,z_] := x^3y^6 fz31[x_,y_,z_] := 2 x^3y^5z fz32[x_,y_,z_] := 3 x^3y^4z^2 fz33[x_,y_,z_] := 4 x^3y^3z^3 fz34[x_,y_,z_] := 5 x^3y^2z^4 fz35[x_,y_,z_] := 6 x^3y z^5 fz36[x_,y_,z_] := 7 x^3z^6 fz37[x_,y_,z_] := 0 fz38[x_,y_,z_] := x^2y^7 fz39[x_,y_,z_] := 2 x^2y^6z fz40[x_,y_,z_] := 3 x^2y^5z^2 fz41[x_,y_,z_] := 4 x^2y^4z^3 fz42[x_,y_,z_] := 5 x^2y^3z^4 fz43[x_,y_,z_] := 6 x^2y^2z^5 fz44[x_,y_,z_] := 7 x^2y z^6 fz45[x_,y_,z_] := 8 x^2z^7 fz46[x_,y_,z_] := 0 fz47[x_,y_,z_] := x y^8 fz48[x_,y_,z_] := 2 x y^7z fz49[x_,y_,z_] := 3 x y^6z^2 fz50[x_,y_,z_] := 4 x y^5z^3 fz51[x_,y_,z_] := 5 x y^4z^4 fz52[x_,y_,z_] := 6 x y^3z^5 fz53[x_,y_,z_] := 7 x y^2z^6 fz54[x_,y_,z_] := 8 x y z^7 fz55[x_,y_,z_] := 9 x z^8 fz56[x_,y_,z_] := 0 fz57[x_,y_,z_] := y^9 fz58[x_,y_,z_] := 2 y^8z fz59[x_,y_,z_] := 3 y^7z^2 fz60[x_,y_,z_] := 4 y^6z^3 fz61[x_,y_,z_] := 5 y^5z^4 fz62[x_,y_,z_] := 6 y^4z^5 fz63[x_,y_,z_] := 7 y^3z^6 fz64[x_,y_,z_] := 8 y^2z^7 fz65[x_,y_,z_] := 9 y z^8 fz66[x_,y_,z_] := 10 z^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]} A43b := {f[2,1,1],fx[2,1,1],fy[2,1,1], f[2,1,-1],fx[2,1,-1],fy[2,1,-1], f[2,-1,1],fx[2,-1,1],fy[2,-1,1], f[-2,1,1],fx[-2,1,1],fy[-2,1,1], f[1,2,1],fx[1,2,1],fy[1,2,1], f[1,2,-1],fx[1,2,-1],fy[1,2,-1], f[1,-2,1],fx[1,-2,1],fy[1,-2,1], f[-1,2,1],fx[-1,2,1],fy[-1,2,1], f[1,1,2],fx[1,1,2],fy[1,1,2], f[1,1,-2],fx[1,1,-2],fy[1,1,-2], f[1,-1,2],fx[1,-1,2],fy[1,-1,2], f[-1,1,2],fx[-1,1,2],fy[-1,1,2], f[1,3,3],fx[1,3,3],fy[1,3,3], f[1,3,-3],fx[1,3,-3],fy[1,3,-3], f[1,-3,3],fx[1,-3,3],fy[1,-3,3], f[-1,3,3],fx[-1,3,3],fy[-1,3,3], f[3,1,3],fx[3,1,3],fy[3,1,3], f[3,1,-3],fx[3,1,-3],fy[3,1,-3], f[3,-1,3],fx[3,-1,3],fy[3,-1,3], f[-3,1,3],fx[-3,1,3],fy[-3,1,3], f[3,3,1],fx[3,3,1],fy[3,3,1], f[3,3,-1],fx[3,3,-1],fy[3,3,-1], f[3,-3,1],fx[3,-3,1],fy[3,-3,1], f[-3,3,1],fx[-3,3,1],fy[-3,3,1], f[1,0,0], fx[1,0,0], fy[1,0,0], f[0,1,0], fx[0,1,0], fy[0,1,0]} Factor[NullSpace[A43b]]