I am motivated by the following paper by Greg Martin and Erick B. Wong:

http://www.math.ubc.ca/~gerg/papers/downloads/AAIMHNIE.pdf

Here the authors prove that assuming that the entries of an $n \times n$ matrix are chosen randomly with respect to a uniform distribution from the set {$-k, -k + 1 \cdots, -1, 0, 1, \cdots, k-1, k$}, then the probability that the resulting matrix will be singular is $\ll k^{-2 + \epsilon}$ (lemma 1 in the above paper).

What I am interested in is a more refined case. Suppose that $H, D$ are positive integers, and suppose that $(a_h, b_h, c_h, d_h)$, $1 \leq h \leq H$, are tuples of integers. Consider the monomials $m_r(a, b, c, d) = a^u b^v c^w d^x$, where $u + v + w + x = D$, and $u,v,w,x$ are non-negative integers. Here we allow $(u,v,w,x)$ to range over all possible choices. Let $R$ be the number of such monomials, which by construction is $\displaystyle \binom{D+3}{3} = \frac{(D+3)(D+2)(D+1)}{6}$, and consider the $H \times R$ matrix where the $hr$-th entry is the monomial $m_r(a_h, b_h, c_h, d_h)$. Finally, we can assume that $R < H$ to make the problem interesting.

Now suppose that the integers $a_h, b_h, c_h, d_h$, $1 \leq h \leq H$ are chosen uniformly from the set {$-k, \cdots, -1, 0, 1, \cdots, k$} say, then what is the probability that the resulting matrix will have rank at most $R-1$?

This amounts to showing that every sub $R \times R$ matrix is singular, and so is related to the original paper cited.