An estimate for the rate of convergence of the distribution of the number of false solutions of a system of nonlinear random equations in the field $GF(2)$

We prove a result on the rate of convergence as $n\to\infty$ of the distribution of the number of false solutions of a system of nonlinear random equations in the field $GF(2)$ to the Poisson distribution with parameter $2^{m}$. We assume, in particular, that the difference between the number $n$ of unknowns and the number $N$ of equations of the system is a constant $m$.