Determinant of a generalized Vandermonde matrix Moore matrix is similar to Vandermonde matrix but has a slightly modified definition.
http://en.wikipedia.org/wiki/Moore_matrix

What is the complexity of computing the determinant of a given $n \times n$ full rank Moore matrix modulo some integer? 

Can Moore determinant be reduced from $O(n^{3})$ using FFT techniques to $O(n\log^{a}n)$ for some $a \in \mathbb{R}_{+} \cup \{0\}$?

Is complexity of Moore det modulo an integer and Vandermonde det the same?
Complexity of Vandermonde determinant is $O(n\log^{2}n)$ (Page 644 in Handbook of Theoretical Computer Science: Algorithms and complexity By Jan Leeuwen)

Post similar to the current one: Determinant modulo m
In general, there is a theoretical $O(n^{2.376})$ time algorithm for finding the LU decomposition of an arbitrary matrix using the Coppersmith–Winograd algorithm, which then obviously yields the determinant (adding $O(n)$ time).  There is a problem however that the Coppersmith–Winograd algorithm isn't considered usable in practice.  Afaik, people mostly use the $O(n^{2.807})$ time Strassen algorithm.  Doesn't Boost UBLAS's lu_factorize use this?

In you case, the Moore matrix should admit considerable optimizations because basically any Gaussian elimination like procedure like LU decomposition can be done abstractly.  Indeed, you'll find a nice $O(n)$ formula for computing the Moore determinant referenced by wikipedia, which presumably one proves by simply working out the LU decomposition in general.