Suppose that $A,B\in M_{p,q}(\mathbb{Z})$ are two rectangular integer matrices of the same size. Suppose that one has a conjecture stating that the rank of the matrix $A+tB$ for Zariski generic values of $t$ is equal to $r$. Is there a way to check that conjecture that is more efficient than the "obvious way" (compute the Smith normal form or Hermite normal form in $\mathbb{Q}[t]$)? Thing is, the "obvious way" does much more: it tells us what values are non-generic, so I am wondering if there is some sneaky way to do less for getting less. (If it helps, the matrices $A$ and $B$ are both sparse and in fact have nonzero elements in exactly the same positions.)
Clarification. By "more efficient" I really mean the practical side: in a problem I am thinking about, the numbers $p$ and $q$ are in hundreds (and there are many pairs $(A,B)$ for which I have a conjecture that needs to be checked), which makes computations unpleasantly slow.
Context: I asked a question about algorithms for Smith normal forms some days ago both on MO and elsewhere, and was (justly) pointed out that simultaneous cross-posting is not appropriate, so I ended up choosing the SciComp as a place to ask it (this is the question). Answering a comment to that question, I realised that the matter of generic rank is what I care to understand the most, and I suspect that if this part of the question can be answered somewhere, that'd be on MathOverflow.
