The Poisson limit theorem for the number of noncollinear solutions of a system of random equations of a special form

We investigate the properties of the number $\nu$ of non-collinear non-zero solutions of a random system of equations of the following form. The left-hand sides of these equations are some functions of linear expressions of the form
$$ l_s=a_{s,1}x_1\oplus\ldots\oplus a_{s,n}x_n $$
with random coefficients and unknowns $x_1,\ldots,x_n$. The right-hand sides are equal to zero. The system is considered over the field $\mathit{GF}(q)$. We assume that the coefficients in $l_s$ are independent and have the uniform distribution. In this paper, we obtain inequalities for the factorial moments of the random variable $\nu$ and give sufficient conditions of validity of the Poisson limit theorem for $\nu$.
The research was supported by the Russian Foundation for Basic Research, grant 99–01–00012, and by the Foundation of the President of the Russian Federation for Support of Scientific Schools, grant 00–15–96136.