  Let $X$ be a matrix with entries in a polynomial ring over an algebraically
closed field $K$. We prove that, if the entries of $X$ outside some $(t \times
t)$-submatrix are algebraically dependent over $K$, the arithmetical rank of
the ideal $I_t(X)$ of $t$-minors of $X$ drops at least by one with respect to
the generic case; under suitable assumptions, it drops at least by $k$ if $X$
has $k$ zero entries. This upper bound turns out to be sharp if
$\mathrm{char}\, K=0$, since it then coincides with the lower bound provided by
the local cohomological dimension.
