d=(z2+3w^2)^2+(x^2+3y^2)(xz+3yw+3xw-3yz)
でなく
d=(z2+3w^2)^2+(x^2+3y^2)(xz+3yw-3xw+3yz)
としてみる.
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a=-(x^2+3y^2)^2+(z^2+3w^2)(xz+3yw+3xw-3yz),
b=(x2+3y^2)^2-(z^2+3w^2)(xz+3yw-3xw+3yz),
c=(z^2+3w^2)^2-(x^2+3y^2)(xz+3yw+3xw-3yz),
d=(z2+3w^2)^2+(x^2+3y^2)(xz+3yw-3xw+3yz)
A=(x^2+3y^2)
B=(z^2+3w^2)
C=(xz+3yw+3xw-3yz)
D=(xz+3yw-3xw+3yz)
E=(xz+3yw-3xw+3yz)とおく.
a=-A^2+BC,
b=A^2-BD,
c=B^2-AC,
d=B^2+AE
a^3=-A^6+3A^4BC-3A^2B^2C^2+B^3C^3
b^3=A^6-3A^4BD+3A^2B^2D^2-B^3D^3
c^3=B^6-3B^4AC+3A^2B^2C^2-A^3C^3
d^3=B^6+3B^4AE+3A^2B^2E^2+A^3E^3
a^3+b^3=3A^4B(C-D)-3A^2B^2(C^2-D^2)+B^3(C^3-D^3)
C-D=6xw-6yz
C+D=2xz+6yw
C^2+CD+D^2=(xz+3yw+3xw-3yz)^2
+(xz+3yw+3xw-3yz)(xz+3yw-3xw+3yz)
+(xz+3yw-3xw+3yz)^2
=2(xz+3yw)^2+2(3xw-3yz)^2
+(xz+3yw)^2-(3xw-3yz)^2
=3(xz+3yw)^2+(3xw-3yz)^2
d^3-c^3=3B^4A(E+C)+3A^2B^2(E^2-C^2)+A^3(E^3+C^3)
E+C=2xz+6yw
E-C=-6xw+6yz
E^2-EC+C^2=(xz+3yw-3xw+3yz)^2-(xz+3yw-3xw+3yz)(xz+3yw+3xw-3yz)+(xz+3yw+3xw-3yz)^2
=2(xz+3yw)^2+2(3xw-3yz)^2
-(xz+3yw)^2+(3xw-3yz)^2
=(xz+3yw)^2+3(3xw-3yz)^2
===================================
F=xz+3yw
G=3xw-3yz
とおくと,
C-D=2G
C+D=2F
C^2+CD+D^2=3F^2+G^2
a^3+b^3=6A^4BG-12A^2B^2FG+2B^3G(3F^2+G^2)
=6A^4BG-12A^2B^2FG+6B^3F^2G+2B^3G^3
F=xz+3yw
G=3xw-3yz
とおくと,
E+C=2xz+6yw=2F
E-C=-6xw+6yz=-2G
E^2-EC+C^2=F^2+3G^2
d^3-c^3=3B^4A(E+C)+3A^2B^2(E^2-C^2)+A^3(E^3+C^3)
=6B^4AF-12A^2B^2FG+2A^3F^3+6A^3FG^3
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