This paper is expository and methodological in nature and is devoted to the development of E.I. Zolotarev’s ideas embedded in his approach to the proof of the quadratic reciprocity law (1872). We consider extensions of Zolotarev’s approach to abstract number rings presented in the work of A. Brunyate and P.L. Clark (2015), and to finite groups studied in the paper by W. Duke and K. Hopkins (2005). We also discuss the connections between these studies and certain classical results, as well as various approaches to proving the quadratic reciprocity law. 

Linear polyelement equations are investigated for functions analytic in a plane with a symmetric cut along the imaginary axis and vanishing at infinity. The solution is sought in the class of functions representable by a Cauchy-type integral over given cuts with an even or odd density. The regularization of these equations and the question of its equivalence are studied. In this process, the theory of elliptic functions is substantially used. The obtained results are applicable to the study of interpolation problems for entire functions of exponential type from class A.

We study the reduced semigroup C^∗-algebras generated by the left regular representations of free products for countable families of rational number semigroups. It is shown that these algebras are nuclear and amenable. We give their characterization as universal C^∗-algebras determined by generators subject to polynomial relations.

We study the interplay between the properties of quantum states on the Hilbert space ℓ₂(κ) and the set-theoretic nature of the cardinal κ. We focus on the existence of singular σ-additive states – functionals whose induced measures are σ-additive yet vanish on singletons. While the existence of such states is known to be equivalent to the Ulam measurability of κ, their structural and dynamical properties remain largely unexplored. We prove that any σ-additive state on the diagonal algebra is representable as a Pettis integral over a singular σ-additive measure, extending the classical representation theory to the non-normal sector. Furthermore, we construct a class of quantum channels using σ-complete ultrafilters that map normal states to singular σ-additive states, effectively «archiving» information into the singular part of the state space.

A class of strictly positive stochastic operators of degree n acting on a one-dimensional simplex is investigated. The main focus is on the problem of the number of fixed points of such operators. It is shown that this problem reduces to studying the number of roots of the corresponding degree n polynomial on the interval (0,1). The existence of at least one fixed point is proved for arbitrary n.

For the quadratic case (n = 2), the uniqueness of the fixed point is established. For a cubic operator (n = 3), conditions for the existence of one, two, or three fixed points are obtained. For a quartic operator (n = 4), criteria determining the number of fixed points are proved in terms of the operator’s parameters.

We consider the isotopic classification problem of arrangements in the real projective plane of two M-curves of degree 4. We study the arrangements that satisfy the maximality condition (an oval of the first curve has 16 pairwise different common points with an oval of the second one) and some combinatorial condition distinguishing a special type. We have obtained a complete classification of algebraic curves of the class under consideration. The proofs of algebraic nonrealizability for the arrangements of the studied class of curves of degree 8 use Orevkov’s method based on braids and links theory.