Are there any results concerning the following problem:

Count all $n \cdot (n + m)$ integer matrices ($m \ge 1$) with norm ($\|A\| = \max{|a_{ij}|}$ or $\|A\| = \sqrt{\sum{a_{ij}^{2}}}$ ) less than or equal to $r$, for which $n \cdot n$ determinants have given values, not all equal to zero.

I'm interested in some asymptotic formula in the variable $r$.