<s>[INST] Rewrite $f(x,y) = 1-x^2y^2$ as a product $g(x) \cdot h(y)$ Rewrite $f(x,y) = 1-x^2y^2$ as a product $g(x) \cdot h(y)$ (both arbitrary functions)
To make more clear what I'm talking about I will give a example.
Rewrite $f(x,y) = 1+x-y-xy$ as $g(x)h(y)$
If we choose $g(x) = (1+x)$ and $h(y) = (1-y)$ we have
$f(x,y) = g(x) h(y) \implies (1+x-y-xy) = (1+x)(1-y)$
I'm trying to do the same with $f(x,y) = 1-x^2y^2 = (1-xy)(1+xy)$.
New question:

Is there also a contradiction for $f(x,y) = \frac{xy}{1-x^2y^2}$ ? Or it's possible to write $f(x,y) $ as $g(x)h(y)$ ?

 [/INST] Without loss of generality let $f(x) = ax^2 + b$ and $g(y) = cy^2 + d$. Now
$$
1-x^2y^2 = f(x)g(y) = acx^2y^2 +adx^2 = cby^2 +bd
$$
By comparing terms we obtain:
\begin{align}
ad&=0\\
cb&=0\\
ac&=-1\\
bd&=1
\end{align}
By $bd=1$ we obtain $a=c=0$. Hence this problem has no solution.
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