This article is concerned with the asymptotic statistics of the number \(\mathcal N^U_N(p^N_1,\dots, p^N_m)\) of zeros in an open set \(U\subset\mathbb C^m\) of a full system \(\{p^N_j\}\) of \(m\) Gaussian random polynomials as the degree \(N\to \infty\). The main result of this article gives an asymptotic formula for the variance of \(\mathcal N^U_N(p^N_1,\dots, p^N_m)\) for open sets with piecewise smooth boundary.