Let $A$ be a finite subset of a field $\mathbb{ F}$ and $D_n(A)$ be a set of all matrices with entries in $A$, namely $$ D_n(A)=\{ D\in \mathbb{ F}\ |\ \exists a_{ ij}\in A, 1 \le i,j \le n, \det\bigl((a_{ ij})\bigr)=D\}, $$ where the symbol $(a_{ ij})$ defines the matrix with elements $a_{ ij}$. How big is the size of the set $D_n(A)$ comparing to the size of the set $A$?
Categories: math.NT