I'm looking at the rank of certain square matrices given some constraints on the entries. The matrix entries are in $\mathbb Z_p$, and I want to minimize the rank under some conditions.
Suppose I know the family of matrices that satisfy my conditions and achieve minimal rank when I compute it over $\mathbb R$. Now I want to do the same over $\mathbb F_p$. Since $\mathbb F_p\not\subseteq\mathbb R$, there is no nice relation that I'm aware of that can directly evaluate the rank exactly. But can I at least say something like:
If $M$ achieves minimal rank over $\mathbb R$, there there is no matrix $M'$ such that $rank_{\mathbb F_p}(M')<rank_{\mathbb F_p}(M)$.
Call the set of matrices in $\mathbb M_n(\mathbb Z_p)$ with rank $C$ over a field $K$, $S_\min(K,n:C)$. What can we say about $|S(\mathbb F_p,n:C)|$ compared to $|S(\mathbb R,n:C)|$? Any pointers towards special cases, or results on well known families of matrices are also helpful!