After answering another question (The number of values of $f(x)/x$ 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 $L(X_1, X_2, ...) = \sum_{\sigma \in S_n} \prod_{i=1}^n Frob^{i-1}(X_{\sigma(i)})$.

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?