On the distribution of the numbers of solutions of random inclusions

For given sets $D$ and $B$ of vectors in linear spaces $V^n$ and $V^T$ over the field $K=GF(q)$ we consider the number of solutions $\xi(D,F,B)$ of the system of inclusions $x\in D$, $A_1x+A_2 f(x)\in B$, where $A_1$ and $A_2$ are random $T\times n$ and $T\times m$ matrices over $K$ with independent elements and $f\colon V^n\to V^m$ is a given mapping. Sufficient conditions for the convergence of distributions of $\xi(D,F,B)$ to the Poisson or compound Poisson distributions are found. Results are applied to the number of solutions of a system of random polynomial equations.