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Multivariate quartic function help

  1. Dec 7, 2009 #1
    1. The problem statement, all variables and given/known data

    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?

    2. Relevant equations

    the multiplication of xy is constant and equal to 1.

    3. 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.
     
    Last edited: Dec 7, 2009
  2. jcsd
  3. Jan 3, 2010 #2
    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.
     
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