Invariance theorems for a class of nonlinear systems of equations over an arbitrary finite field
Consider the following system of equations in the field \(\text{GF}(2)\): \[ \sum^{d_i}_{k= 1} \sum_{1\leq j_1<\cdots< j_k\leq n} a_{i,j_1,\dots,j_k} x_{j_1}\cdots x_{j_k}= b_i,\quad i=1,\dots,n- s, \] where \(s\) is a fixed integer, \(a_{i,j_1,\dots,j_k}\) are mutually independent random variables, \(b= (b_1,\dots, b_{n-s})\) is obtained by replacing the given vector \(x^0= (x^0_1,\dots, x^0_n)\) into the system of equations, and \(d_i\) are natural numbers such that \(2\leq d_i\leq n\), \(i=1,\dots,n- s\). Let \(\nu_n\) denote the number of solutions different from \(x^0\). The author studies the asymptotic behavior of \(\nu_n\) as \(n\to\infty\) in terms of the limiting distribution and describes the geometrical structure of the set of solutions of the system provided \(x^0= (0,\dots,0)\). The results are also extended to the finite field \(\text{GF}(q)\) for an arbitrary \(q\geq 2\).
equations over an arbitrary finite field
scientific article; zbMATH DE number 1051117