My task is to simplify at most the following polynomial $f \in\mathbb{Q}[x_1,x_2,x_3,x_4]$ (I must then write it as alementary symmetric polynomial)
$f:=\frac{\det\begin{pmatrix}1&x_1^2&x_1^3&x_1^4\\1&x_2^2&x_2^3&x_2^4\\1&x_3^2&x_3^3&x_3^4\\1&x_4^2&x_4^3&x_4^4 \end{pmatrix}}{\det\begin{pmatrix}1&x_1&x_1^2&x_1^3\\1&x_2&x_2^2&x_2^3\\ 1&x_3&x_3^2&x_3^3\\1&x_4&x_4^2&x_4^3 \end{pmatrix}}$
Clearly the denominator is equal to $\prod_{1≤i<j≤4}(x_j-x_i)=(x_4-x_1)(x_4-x_2)(x_4-x_3)(x_3-x_2)(x_3-x_1)(x_2-x_1)$ being a Vandermonde determinant.
The problem is the numerator, I tried to compute it, but even after a long computation i can't see an easy way to determine the fraction. I was wondering if also the numerator is correlated with a Vandermonde matrix somehow, or if there is something I'm missing. Thank you in advance.
Consider the polynomial $$P(T)=(T-x_1)(T-x_2)(T-x_3)(T-x_4)\tag{1}$$
Then $P$ can be rewritten as
$$ P(T)=T^4-\sigma_1T^3+\sigma_2T^2-\sigma_3T+\sigma_4 \tag{2} $$
where the $\sigma_k$ are the elementary symmetric polynomials. Now by (1), we have $P(x_k)=0$ for every $k$, so (2) yields
$$ x_k^4 = \sigma_1x_k^3-\sigma_2x_k^2+\sigma_3x_k-\sigma_4 \tag{3} $$
Using (3), we can rewrite each entry in the last column of the numerator of $f$. Using operations on columns, we then easily deduce $f=-\sigma_3$.