519.21The results on systems of random equations over finite algebraic structures are reviewed. Basic definitions, concepts, and problems in this field are presented.Keywords: system of random equations over finite algebraic structure (finite field, finite ring, finite Abelian group), Boolean system of equations, certainly compatible system of random equations, system of random equations with independent left-and right-hand sides, system of random equations with distorted right-hand sides.This review of the results from the studies on systems of random equations over finite algebraic structures is timed to the 70th anniversary of Igor N. Kovalenko, a prominent Ukrainian mathematician, Doctor of engineering, physics, and mathematics, Academician of the National Academy of Sciences of Ukraine. He may rightfully be considered a founder of this complex and fascinating division in discrete mathematics. His fundamental and mathematically beautiful study [33] (1967) on the invariance of the limit behavior of the number of solutions of one class of systems of random equations over a GF(2) field became a basis for disciples of I. N. Kovalenko's school in extending invariance theory to random systems over arbitrary finite fields and finite rings. Over nearly 40 years since the paper [33] was published, the theory of systems of random equations over finite algebraic structures has been enriched with new problems, results, and methods. To efficiently develop this theory, we today need to sort out and systematize everything accumulated in this area. The author of the present review, a follower of I. N. Kovalenko, has tried to do this with gratitude and deep respect for the teacher.The studies by V. L. Goncharov on random substitutions and series in random (0,1)-sequences, published in the 40s of the last century, have initiated a systematic analysis of combinatory objects using methods of probability theory. Thus, they promoted the development of a new division in discrete mathematics, probabilistic combinatorial analysis, where the theory of systems of random equations over finite algebraic structures is a relatively small subdivision, despite its important applications in the theory of finite state automata, the theory of coding, the theory of random graphs, cryptography, etc. there is an explanation of this. The use of random systems as probability-theoretic models of combinatorial processes occurring in information-protection systems has vetoed publications on this subject in the available scientific literature for many years. The situation began to change for the better only in the early 90s of the last century. Indeed, a large number of papers on random systems over finite algebraic structures have appeared in the last 10 to 14 years in large mathematical journals. It is pleasant to point out that whereas this subjects was considered especially "Russian" when the review [40,41] devoted to random linear systems over finite algebraic structures was written (since the corresponding publications were mainly due to Soviet ma...

519.21A class of homogenous systems of random nonlinear equations over an arbitrary finite ring with left unity is considered. The author analyzes the invariance boundaries for limit factorial moments of nonzero solutions, the limit distribution of the number of nonzero solutions, and the geometrical structure of the set of nonzero solutions of the system as the number of unknowns tends to infinity.Keywords: system of random nonlinear (linear) equations over a finite ring with left unity (over a finite field), factorial moment of the number of nonzero solutions of a system, distribution of the number of nonzero solutions of a system.

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