After answering another question ([The number of values of $f(x)/x$ when $f$ is a linearized polynomial](https://mathoverflow.net/questions/268339/the-number-of-values-of-fx-x-when-f-is-a-linearized-polynomial)), I stumbled upon an interesting polynomial in multiple variables. Let $\mathbb{F}\_q$ be the field of $q$ elements, and let $K$ be a field containing it. Then define 
\begin{equation}
L(X\_1, X\_2, ...) = \sum\_{\sigma \in S\_n} \text{sgn}(\sigma) \prod\_{i=1}^n \text{Frob}^{i-1}(X\_{\sigma(i)}).
\end{equation}
This polynomial is $\mathbb{F}\_q$-multilinear, and detects whether the elements are linearly dependent over $\mathbb{F}\_q$. In other words, it is nonzero exactly when they are linearly independent. I think this can be extended to the case where $K$ is an algebra, but haven't checked. 


This seems interesting enough that I am guessing it's been found and used before; does this polynomial have a name, and is it used anywhere interesting?



A polynomial in multiple variables with nice properties
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](https://en.wikipedia.org/wiki/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 $\mathbb{F}\_q \mathbb{P}^{n-1}$.


