Fedor Pakovich

Solution of the polynomial moment problem

In this paper we give a solution of the following ‘polynomial moment problem’ which arose about

A counterexample to the “composition conjecture"

In this note we construct a class of counterexamples to the composition conjecture concerning an infinitesimal version of the center problem for the polynomial Abel equation in the complex domain.

Cauchy Type Integrals of Algebraic Functions

AbstractWe consider Cauchy-type integrals

SOLUTION OF THE HURWITZ PROBLEM FOR LAURENT POLYNOMIALS

I(t) = \frac{1}{{2\pi i}} \int_\gamma {\frac{{g(z)dz}}{{z - t}}}

On semiconjugate rational functions

withg(z) an algebraic function. The main goal is to give constructive (at least, in principle) conditions forI(t) to be an algebraic function, a rational function, and ultimately an identical zero near infinity. This is done by relating the monodromy group of the algebraic functiong, the geometry of the integration curve γ, and the analytic properties of the Cauchy-type integrals. The motivation for the study of these conditions is provided by the fact that certain Cauchy-type integrals of algebraic functions appear in the infinitesimal versions of two classical open questions in Analytic Theory of Differential Equations: the Poincaré Center-Focus problem and the second part of Hilbert’s 16-th problem.

Algebraic curves P(x) − Q(y) = 0 and functional equations

In this paper we prove several results about the lattice of imprimitivity systems of a permutation group containing a cyclic subgroup with at most two orbits. As an application we generalize the first Ritt theorem about functional decompositions of polynomials, and some other related results. Besides, we discuss examples of rational functions, related to finite subgroups of Aut(CP 1 ), for which the first Ritt theorem fails to be true.

On the functional equation F(A(z)) = G(B(z)), where A,B are polynomials and F,G are continuous functions

We investigate the following existence problem for rational functions: for a given collection Π of partitions of a number n to define whether there exists a rational function f of degree n for which Π is the branch datum. An important particular case when the answer is known is the one when the collection Π contains a partition consisting of a single element (in this case, the corresponding rational function is equivalent to a polynomial). In this paper, we provide a solution in the case when Π contains a partition consisting of two elements.

On decompositions of trigonometric polynomials

In this paper we describe polynomials orthogonal to all powers of a Chebyshev polynomial on a segment.