Good evening all, I am determined to determine this determinant:
$D = \det{\left[x_j^{n-i} - x_j^{2n-i}\right]_{i,j=1}^{n}}$
Looking at the smaller cases, leads me to believe that
$D = \prod_{1 \leq i < j \leq n}\left(x_i-x_j\right)\prod_{i=1}^n \left(1-{x_i}^n\right)$
although I am having trouble showing this. I know that, since the determinant is an alternating function in the variables $x_1,\dots x_n$ it follows that
$ \frac{D}{\displaystyle\prod_{1 \leq i < j \leq n}\left(x_i-x_j\right)} $
is a symmetric polynomial of degree $n^2$ (the degree of D minus the degree of the Vandermonde part).
How can I show that this symmetric polynomial is exactly $\prod_{i=1}^n \left(1-{x_i}^n\right)$ ?
Your help is, as always, much appreciated.