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I am trying to find the solutions to the following system over the complex numbers:

[tex] x^4 + y^4 + z^4 +4xyz - 4x^2 - 4y^2 - 4z^2 +5 =0[/tex]

[tex] 4x^3 + 4yz - 8x =0[/tex]

[tex] 4y^3 + 4xz - 8y =0[/tex]

[tex] 4z^3 + 4xy - 8z =0[/tex]

I calculated the resultant:

[tex] xyz - 2x^2 - 2y^2 - 2z^2 +5 [/tex]

which would give a necessary condition for solutions but I've been messing about for hours and I still can't see how to proceed.

Any suggestions?

[tex] x^4 + y^4 + z^4 +4xyz - 4x^2 - 4y^2 - 4z^2 +5 =0[/tex]

[tex] 4x^3 + 4yz - 8x =0[/tex]

[tex] 4y^3 + 4xz - 8y =0[/tex]

[tex] 4z^3 + 4xy - 8z =0[/tex]

I calculated the resultant:

[tex] xyz - 2x^2 - 2y^2 - 2z^2 +5 [/tex]

which would give a necessary condition for solutions but I've been messing about for hours and I still can't see how to proceed.

Any suggestions?

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