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 $r$.
This seems to be a deep question on (diophantine) geometry of numbers. There are papers by Margulis and others estimating asymptotically the number of integral points on surfaces given by polynomial equations in $\mathbb R^n$.
Even for a hyperplane it leads to questions of simultanious approximation of numbers (the coefficients of the equation) by rationals which have deep answers.
