On determinantal ideals and algebraic dependence

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×t)-submatrix are algebraically dependent over K, the arithmetical rank of the ideal It(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 char K=0, since it then coincides with the lower bound provided by the local cohomological dimension.