MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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?

share|cite|improve this question
up vote 1 down vote accepted

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.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.