I'm currently trying to understand the paper "Laurent Determinants and Arrangements of Hyperplane Amoebas" by Forsberg, Passare and Tsikh, which can be found here https://core.ac.uk/download/pdf/82749806.pdf.
Early on in the paper, they introduce the following map, which they define to be the order of a point $\mathbf{u} = (u_1, \dots u_n)$ of the amoeba complement:
$$v_j( \mathbf{u} ) = \frac{1}{(2 \pi i)^n} \int_{log^{-1}(\mathbf{x})} \ \left( \frac{z_j \hspace{0.7mm} \delta_j f(\mathbf{z}) }{ f(\mathbf{z}) \ z_1 \cdots z_n } \right) d z_1 \cdots d z_n \ .$$
They then make the following argument to prove that $v_j$ is integer valued:
Write the coordinates in polar form $z_k=exp(u_k+i\theta_k)$, and consider, for fixed arguments $\theta_k$, $k \neq j$ the ordinary contour integral
$$\frac{1}{(2 \pi i)} \int_{ \left| z_J\right| = e^{u_j}} \ \frac{ \delta_1 f(z_1) }{ f(z_1 ) } d z_j \ . $$
By the classical argument principle this integral will be integer valued. Since it also depends continuously on the remaining arguments, it must in fact be independent of these, and it follows that its value is equal to $v_j$
I don't think I really understand why this is the case, however - my guess is that varying the arguments $\theta_j$ will not affect the number of zeroes and poles of $f(z_1)$, hence it will not change the single variable contour integral. However I don't understand why this will not then change the multiple variable integral $v_j$. Any help would be greatly appreciated.
Regards,
b_dobres