## 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 $$GF(2)$$.(Ukrainian, English)Zbl 1199.60030

Teor. Jmovirn. Mat. Stat. 77, 109-121 (2007); translation in Theory Probab. Math. Stat. 77, 121-134 (2008).
The authors deal with the system of equations
$\sum_{k=1}^{g_i(n)}\sum_{1\leq j_1<\cdots<j_k\leq n}a_{j_1,\dots,j_k}^{(i)} x_{j_1}\cdots x_{j_k}=b_i,\,i=1,\dots,N,$
over the field $$GF(2)$$ containing only two elements where the coefficients $$a_{j_1,\dots,j_k}^{(i)}, 1\leq j_1<\cdots<j_k\leq n,k=1,\dots,g_i(n)$$ are independent random variables taking the value 1 with probability $$P\{a_{j_1,\dots,j_k}^{(i)}=1\}=p_{ik}$$, and the value 0 with probability $$1-p_{ik}$$, the elements $$b_i,i=1,\dots,N$$, are equal to the left hand side of the system, if a fixed $$n$$-dimensional vector $$\vec{x}_0$$ is substituted for unknowns, and the functions $$g_i(n),i=1,\dots,N$$, are nonrandom and such that $$g_i(n)\in \{2,\dots,N\}$$.
Denote by $$\nu(n)$$ the number of false solutions of the system, that is, the number of solutions that do not coincide with the vector $$\vec{x}_0$$. The rate of convergence of the distribution of the random variable $$\nu(n)$$ to the limit Poisson distribution as $$n\to\infty$$ is estimated.
For more results and references see [V. G. Mikhailov, Theory Probab. Appl. 43, No. 3, 480–487 (1998; Zbl 0951.60011); translation from Teor. Veroyatn. Primen. 43, No. 3, 598–606 (1998)].

### MSC:

 60C05 Combinatorial probability 15B52 Random matrices (algebraic aspects) 15A03 Vector spaces, linear dependence, rank, lineability

Zbl 0951.60011
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