Multivariate quartic function help

[rattam]

Homework Statement

I am a Turkish student bogged down by a very difficult question (at least for me).

Here's the question:

Given that xy= 1, find all solutions for X4 + y4 - z4 + 4z2 - 2 = 0.

I've spent many hours on this equation. It even prevents me from sleep.

I would appreciate it deeply if someone were to solve it for me, but I'll be equally grateful being directed to a book where equations of this kind are handled. Is this a trivial question? Is it simple to answer with the necessary background?

Homework Equations

the multiplication of xy is constant and equal to 1.

The Attempt at a Solution

I have tried various ways of expanding it, tried to rewrite the z part as binomial, but I think this is just beyond my skill.

Thanks so much for pputting up with my desperation! Any response is appreciated very much.

 

[NSAC]

I would say the solution is some kind of a line on the xy=1 plane in 3D. Your solution would be a closed form of an equation. So I don't know if saying that
{(x,1/x,z) satisfying the equation x^4+1/x^4-z^4-4z^2-2=0 }
is the solution set is enough.