arXiv subjects

F. Pakovich

Publications and source records attributed to F. Pakovich.

24 records · Page 2Linked to original sources

arXiv2007

Solution of the Hurwitz problem for Laurent polynomials

In this paper 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 to this problem 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.

math.GT↗

arXiv2007

A remark on the Chebotarev theorem about roots of unity

Let $Ω$ be a matrix with entries $a_{i,j}=ω^{ij},$ $1\leq i,j \leq n,$ where $ω=e^{2π\sqrt{-1}/n},$ $n\in \mathbb N.$ The Chebotarev theorem states that if $n$ is a prime then any minor of $Ω$ is non-zero. In this note we provide an analogue of this statement for composite $n.$

math.NT↗

arXiv2006

On polynomials sharing preimages of compact sets and related questions

In this paper we give a solution of the following problem: under what conditions on infinite compact sets $K_1,K_2\subset \C$ and polynomials $f_1,$ $f_2$ the preimages $f_1^{-1}\{K_1\}$ and $f_2^{-1}\{K_2\}$ coincide. Besides, we investigate some related questions. In particular, we show that polynomials sharing an invariant compact set distinct from a point have equal Julia sets.

math.DS↗

arXiv2003

Cauchy Type Integrals of Algebraic Functions

We consider Cauchy type integrals $I(t)={1\over 2πi}\int_γ {g(z)dz\over z-t}$ with $g(z)$ an algebraic function. The main goal is to give constructive (at least, in principle) conditions for $I(t)$ 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 function $g$, 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 the Hilbert 16-th problem.

math.CA↗

arXiv2002

On the polynomial moment problem

We treat the following "polynomial moment problem": for a complex polynomial P(z) and distinct complex numbers a,b such that P(a)=P(b) to describe polynomials q(z)=Q'(z) orthogonal to all degrees of P(z) on the segment [a,b]. We show that if P(z) is indecomposable then there exists a polynomial R(z) such that Q(z)=R(P(z)). We also prove some related results.

math.CV↗

arXiv2002

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.

math.DS↗