I want to answer the following question.
Compute the pole set of the rational function $\zeta=\frac{x_{1}+x_{2}}{x_{3}+x_{4}}$ defined on an algebraic variety $V=\mathrm{V}(X_{1}^{2}-X_{2}^{2}+X_{3}^{2}-X_{4}^{2})\subseteq A^{4}(\mathbb{C}).$
I did the following reasoning:
The pole set is $\{(x_1, x_2, x_3, x_4)\in V \text{ s. t. } x_3+x_4=0 \}$. This leads to $x_4=-x_3$, therefore $x_1^2-x_2^2+x_3^2-x_3^2=x_1^2-x_2^2=0$. So $x_1^2=x_2^2$.
Then the pole set is generated by $(x_1, \sqrt {x_1^2}, x_3, -x_3)$
Usually this kind of exercises are longer and I think that maybe I am forgetting something because it was too easy. Is this answer true?