Return to Revisions

This is a CW answer to remove this question from the unanswered list (once someone upvotes it). This is the determinant of the Moore matrix $\left( x_i^{q^{j-1}} \right)_{1 \leq i,j \leq n}$. This determinant can be expressed as a product of linear factors: $$\det \left(x_i^{q^{j-1}} \right) = \prod_{(c_1:c_2:\cdots:c_n) \in \mathbb{F}_q \mathbb{P}^{n-1}} (c_1 x_1 + \cdots + c_n x_n),$$ up to a scalar factor depending on how we choose the representatives $(c_1, \ldots, c_n)\in \mathbb{F}_q^n$ of the points of $\mathbb{F}_q \mathbb{P}^{n-1}$.