Linear independence in the algebraic closure of $\mathbb{C}(z)$

Question

Fix $N>0$. Let $b_i=(b_{i,1}, b_{i,2}, b_{i,3}, b_{i,4})$, $i=1,\ldots, m$, be distinct 4-tuples of integers with with all $0\leq b_{i,j}< N$. (The zero tuple is disallowed.)

Define $w_i=(\prod_{j=1}^4 (z-z_j)^{b_{i,j}})^{\frac1{N}}$.

Consider $w_i$ as an element of the following vector space: the algebraic closure of $\mathbb{C}(z)$ over the field $\mathbb{C}(z)$.

I believe that Puiseux series can be used to show the linear independence of the $w_i$. Are there any other approaches that might show the linear independence of the $w_i$?

More generally, are there any other general techniques for proving linear independence of functions in the given vector space?

Answer 1

You can also use Galois theory or monodromy. Take a minimal linear dependence relation and apply the automorphimsm of the algebraic closure that fixes $(z-z_j)^{1/N},j>1$ and fot $j=1$ multiplies the function by an N-th root of unity, thus getting a new relation and you can produce a shorter relation from those two.