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The Moore matrix over $\mathbb{F}_q$ is the $n\times n$ matrix whose i'th row is: $a_i,a_i^q,a_i^{q^2},\dots,a_i^{q^{n-1}}$. The determinant of this matrix is the product of all linear combinations (over $\mathbb{F}_q$) of the $a_i$'s, and so is zero iff they are linearly dependent. (see: Moore matrix)

Is anything known about the case where the powers in the matrix are non-consecutive powers of q? For instance the matrix whose rows are: $a_i,a_i^q,a_i^{q^2},\dots,a_i^{q^{n-3}},a_i^{q^{n-1}},a_i^{q^{n+1}}$. If we consider the determinant as a polynomial in the $a_i$'s, then this determinant is divisible by the regular Moore determinant, but is there anything more that can be said about it?

Perhaps more specifically, I'm interested in what conditions on the $a_i$'s imply the singularity or non-singularity of this matrix, even just for $q=2$.

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