Geodesic
On the distribution of the numbers of solutions of random inclusions
Matematičeskie voprosy kriptografii, Tome 2 (2011) no. 2, pp. 55-80 
V. A. Kopytcev; V. G. Mikhailov. On the distribution of the numbers of solutions of random inclusions. Matematičeskie voprosy kriptografii, Tome 2 (2011) no. 2, pp. 55-80. http://geodesic.mathdoc.fr/item/MVK_2011_2_2_a2/
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     author = {V. A. Kopytcev and V. G. Mikhailov},
     title = {On the distribution of the numbers of solutions of random inclusions},
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     url = {http://geodesic.mathdoc.fr/item/MVK_2011_2_2_a2/}
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Voir la notice de l'article provenant de la source Math-Net.Ru

For given sets $D$ and $B$ of vectors in linear spaces $V^n$ and $V^T$ over the field $K=GF(q)$ we consider the number of solutions $\xi(D,F,B)$ of the system of inclusions $x\in D$, $A_1x+A_2 f(x)\in B$, where $A_1$ and $A_2$ are random $T\times n$ and $T\times m$ matrices over $K$ with independent elements and $f\colon V^n\to V^m$ is a given mapping. Sufficient conditions for the convergence of distributions of $\xi(D,F,B)$ to the Poisson or compound Poisson distributions are found. Results are applied to the number of solutions of a system of random polynomial equations.

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