The number of solutions of a random system of polynomials over a finite field

We study the probability distribution of the number of common zeros of a system of \(m\) random \(n\)-variate polynomials over a finite commutative ring \(R\). We compute the expected number of common zeros of a system of polynomials over \(R\). Then, in the case that \(R\) is a field, under a necessary-and-sufficient condition on the sample space, we show that the number of common zeros is binomially distributed.