I have the following problem:
Given the polynomial $$ P_n (x_1, ..., x_k) = x_1 ^ n + \cdots + x_k ^ n $$ I would like to give an explicit formula (that depends on $ k, n $) to write to $ P_n (x_1, ..., x_k) $ in terms of $ \sigma_1 (x_1, ..., x_k), ..., \sigma_n (x_1, ..., x_k) $ (with $\sigma_i (x_1 ,. .., x_n)$ the $i$th-elementary symmetric polynomial on $ x_1, ..., x_n $), that is, I would like to find a formula for $ s_n $, where $$ P_n (x_1, ..., x_k) = s_n (\sigma_1 (x_1, ..., x_k), ..., \sigma_n (x_1, ..., x_k)) $$
For example:
- $s_1 (x_1,...,x_k) = \sigma_1 (x_1,...,x_n)$
- $s_2 (x_1,...,x_k) = \sigma_1^2 (x_1,...,x_n) - 2\sigma_2 (x_1,...,x_n)$
- $s_3 (x_1,...,x_k) = \sigma_1^3 (x_1,...,x_n) +3\sigma_3 (x_1,...,x_n) - 3 \sigma_1 (x_1,...,x_n) \sigma_2 (x_1,...,x_n)$
Does anyone know how to attack this problem or have any hint?
I already tried induction and recursion but I didn't get anywhere, I don't know if my calculations are wrong