<s>[INST] Equation $1+x^8y^4+x^4y^8-x^2y^4-x^6y^6-x^4y^2=0$ How to prove that the following equation: $$1+x^8y^4+x^4y^8-x^2y^4-x^6y^6-x^4y^2=0$$
has for solution(in real numbers): $|x|=|y|=1~$ only.
Any hint would be appreciated.
 [/INST] Note that the proposed equation is equivalent to
$$
(1-x^2y^4)^2+(1-x^4y^2)^2+x^4y^4(x^2-y^2)^2=0
$$
So, any solution satisfies $x^2y^4=x^4y^2=1$ and $xy(x^2-y^2)=0$. This implies clearly that $x^2=y^2=1$.$\qquad\square$
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