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F. Pakovich

Publications and source records attributed to F. Pakovich.

At least 19 recordsLinked to original sources
arXiv2020

Finiteness theorems for commuting and semiconjugate rational functions

Let $B$ be a fixed rational function of one complex variable of degree at least two. In this paper, we study solutions of the functional equation $A\circ X=X\circ B$ in rational functions $A$ and $X$. Our main result states that, unless $B$ is a Lattès map or is conjugate to $z^{\pm d}$ or $\pm T_d$, the set of solutions is finite, up to some natural transformations. In more detail, we show that there exist finitely many rational functions $A_1, A_2,\dots, A_r$ and $X_1, X_2,\dots, X_r$ such that the equality $A\circ X=X\circ B$ holds if and only if there exists a Möbius transformation $μ$ such that $A=μ\circ A_j\circ μ^{-1}$ and $X=μ\circ X_j\circ B^{\circ k}$ for some $j,$ $1\leq j \leq r,$ and $k\geq 1$. We also show that the number $r$ and the degrees $°X_j,$ $1\leq j \leq r,$ can be bounded from above in terms of the degree of $B$ only. As an application, we prove an effective version of the classical theorem of Ritt about commuting rational functions.

math.DS↗
arXiv2018

Parametric Center-Focus Problem for Abel Equation

The Abel differential equation $y'=p(x)y^3 + q(x) y^2$ with meromorphic coefficients $p,q$ is said to have a center on $[a,b]$ if all its solutions, with the initial value $y(a)$ small enough, satisfy the condition $y(a)=y(b)$. The problem of giving conditions on $(p,q,a,b)$ implying a center for the Abel equation is analogous to the classical Poincaré Center-Focus problem for plane vector fields. Following [3,4,8,9] we say that Abel equation has a "parametric center" if for each $\varepsilon \in \mathbb C$ the equation $y'=p(x)y^3 + \varepsilon q(x) y^2$ has a center. In the present paper we use recent results of [15,6} to show show that for a polynomial Abel equation parametric center implies strong "composition" restriction on $p$ and $q$. In particular, we show that for $°p,q \leq 10$ parametric center is equivalent to the so-called "Composition Condition" (CC) on $p,q$. Second, we study trigonometric Abel equation, and provide a series of examples, generalizing a recent remarkable example given in [8], where certain moments of $p,q$ vanish while (CC) is violated.

math.CA↗
arXiv2018

Semiconjugate rational functions: a dynamical approach

Using dynamical methods we give a new proof of the theorem saying that if $A,B,X$ are rational functions of degree at least two such that $A\circ X=X\circ B$ and $\mathbb C(B,X)=\mathbb C(z)$, then the Galois closure of the field extension $\mathbb C(z)/\mathbb C(X)$ has genus zero or one.

math.DS↗
arXiv2017

On decompositions of trigonometric polynomials

Let $\mathbb R_t[θ]$ be the ring generated over $\mathbb R$ by $\cosθ$ and $\sinθ$, and $\mathbb R_t(θ)$ be its quotient field. In this paper we study the ways in which an element p of $\mathbb R_t[θ]$ can be decomposed into a composition of functions of the form $p=R(q),$ where $\mathbb R\in \mathbb R(x)$ and $q\in \mathbb R_t(θ)$. In particular, we describe all possible solutions of the functional equation $R_1(q_1)=R_2(q_2)$, where $R_1, R_2 \in \mathbb R[x]$ and $q_1,q_2\in \mathbb R_t[θ].$

math.CA↗
arXiv2016

On semiconjugate rational functions

We investigate semiconjugate rational functions, that is rational functions $A,$ $B$ related by the functional equation $A\circ X=X\circ B$, where $X$ is a rational function of degree at least two. We show that if $A$ and $B$ is a pair of such functions, then either $B$ can be obtained from $A$ by a certain iterative process, or $A$ and $B$ can be described in terms of orbifolds of non-negative Euler characteristic on the Riemann sphere.

math.DS↗
arXiv2014

Algebraic Geometry of the Center-Focus problem for Abel Differential Equation

The Abel differential equation $y'=p(x)y^3 + q(x) y^2$ with polynomial coefficients $p,q$ is said to have a center on $[a,b]$ if all its solutions, with the initial value $y(a)$ small enough, satisfy the condition $y(a)=y(b)$. The problem of giving conditions on $(p,q,a,b)$ implying a center for the Abel equation is analogous to the classical Poincaré Center-Focus problem for plane vector fields. Center conditions are provided by an infinite system of "Center Equations". An important new information on these equations has been obtained via a detailed analysis of two related structures: Composition Algebra and Moment Equations (first order approximation of the Center ones). Recently one of the basic open questions in this direction - the "Polynomial moments problem" - has been completely settled in \cite{mp1,pak}. In this paper we present a progress in the following two main directions: First, we translate the results of \cite{mp1,pak} into the language of Algebraic Geometry of the Center Equations. On this base we obtain new information on the center conditions, significantly extending, in particular, the results of \cite{broy}. Second, we study the "second Melnikov coefficients" (second order approximation of the Center equations) showing that in many cases vanishing of the moments and of these coefficients is sufficient in order to completely characterize centers.

math.CA↗
arXiv2013

Minimum Degree of the Difference of Two Polynomials over Q, and Weighted Plane Trees

A weighted bicolored plane tree is a bicolored plane tree whose edges are endowed with positive integral weights. The degree of a vertex is defined as the sum of the weights of the edges incident to this vertex. Using the theory of dessins d'enfants, which studies the action of the absolute Galois group on graphs embedded into Riemann surfaces, we show that a weighted plane tree is a graphical representation of a pair of coprime complex polynomials A,B such that: (a) deg A = deg B, and A and B have the same leading coefficient; (b) the multiplicities of the roots of A (respectively, of B) are equal to the degrees of the black (respectively, white) vertices of the corresponding tree; (c) the degree of the difference A-B attains the minimum which is possible for the given multiplicities of the roots of A and B. Moreover, if a tree in question is uniquely determined by the set of its black and white vertex degrees (we call such trees unitrees), then the corresponding polynomials are defined over Q. The pairs of polynomials A,B such that the degree of the difference A-B attains the minimum, and especially those defined over Q, are related to some important questions of number theory. Dozens of papers were dedicated to their study. The main result of this paper is a complete classification of the unitrees which provides us with the most massive class of such pairs defined over Q. We also study combinatorial invariants of the Galois action on trees, as well as on the corresponding polynomial pairs, which permit us to find yet more examples defined over Q. In a subsequent paper we compute the polynomials A,B corresponding to all the unitrees.

math.NT↗
arXiv2011

Moments on Riemann surfaces and hyperelliptic Abelian integrals

In the present paper we solve the following different but interrelated problems: (a) the moment problem on Riemann surfaces, (b) the vanishing problem of polynomial Abelian integrals of dimension zero on the projective plane, (c) the vanishing problem of polynomial hyperelliptic Abelian integrals.

math.DS↗
arXiv2010

Generalized "second Ritt theorem" and explicit solution of the polynomial moment problem

In the recent paper arXiv:0710.4085 was shown that any solution of "the polynomial moment problem", which asks to describe polynomials Q orthogonal to all powers of a given polynomial P on a segment, may be obtained as a sum of some "reducible" solutions related to different decompositions of P into a composition of two polynomials of lesser degrees. However, the methods of arXiv:0710.4085 do not permit to estimate the number of necessary reducible solutions or to describe them explicitly. In this paper we provide a description of the polynomial solutions of the functional equation P=P_1(W_1)=P_2(W_2)=...=P_r(W_r), and on this base describe solutions of the polynomial moment problem in an explicit form suitable for applications. With respect to the previous version a more general form of the generalized "secon Ritt theorem" is proved and the proof is considerably simplified. Besides, a missed case in Theorem 1.2 was added and the proof is corrected.

math.DS↗
arXiv2009

Laurent polynomial moment problem: a case study

In recent years, the so-called polynomial moment problem, motivated by the classical Poincare center-focus problem, was thoroughly studied, and the answers to the main questions have been found. The study of a similar problem for rational functions is still at its very beginning. In this paper, we make certain progress in this direction; namely, we construct an example of a Laurent polynomial for which the solutions of the corresponding moment problem behave in a significantly more complicated way than it would be possible for a polynomial.

math.CV↗
arXiv2009

On rational functions orthogonal to all powers of a given rational function on a curve

In this paper we study the generating function f(t) for the sequence of the moments \int_γP^i(z)q(z)d z, i\geq 0, where P(z),q(z) are rational functions of one complex variable and γis a curve in C. We calculate an analytical expression for f(t) and provide conditions implying the rationality and the vanishing of f(t). In particular, for P(z) in generic position we give an explicit criterion for a function q(z) to be orthogonal to all powers of P(z). Besides, we prove a stronger form of the Wermer theorem, describing analytic functions satisfying \int_{S^1}h^i(z)g^j(z)g'(z)d z=0, i\geq 0, j\geq 0, in the case where the functions h(z),g(z) are rational. We also generalize the theorem of Duistermaat and van der Kallen about Laurent polynomials L(z) whose integral positive powers have no constant term, and prove other results about Laurent polynomials L(z),m(z) satisfying \int_{S^1}L^i(z)m(z)d z=0, i\geq i_0.

math.CV↗
arXiv2009

On the equation P(f)=Q(g), where P,Q are polynomials and f,g are entire functions

In 1922 Ritt described polynomial solutions of the functional equation P(f)=Q(g). In this paper we describe solutions of the equation above in the case when P,Q are polynomials while f,g are allowed to be arbitrary entire functions. In fact, we describe solutions of the more general functional equation s=P(f)=Q(g), where s,f,g are entire functions and P,Q are arbitrary rational functions. Besides, we solve the problem of description of "strong uniqueness polynomials" for entire functions.

math.CV↗
arXiv2009

Solution of the polynomial moment problem

In this paper we give a complete solution of the following "polynomial moment problem" which arose about 10 years ago in connection with Poincare's center-focus problem. For a given polynomial P(z) to describe polynomials Q(z) orthogonal to all powers of P(z) on a segment [a,b].

math.CV↗
arXiv2009

Jordan-Holder theorem for imprimitivity systems and maximal decompositions of rational functions

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 the automorphism group of the sphere for which the first Ritt theorem fails to be true.

math.CV↗
arXiv2008

Prime and composite Laurent polynomials

In 1922 Ritt constructed the theory of functional decompositions of polynomials with complex coefficients. In particular, he described explicitly indecomposable polynomial solutions of the functional equation f(p(z))=g(q(z)). In this paper we study the equation above in the case when f,g,p,q are holomorphic functions on compact Riemann surfaces. We also construct a self-contained theory of functional decompositions of rational functions with at most two poles generalizing the Ritt theory. In particular, we give new proofs of the theorems of Ritt and of the theorem of Bilu and Tichy.

math.CV↗
arXiv2008

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

In this paper we give several conditions implying the irreducibility of the algebraic curve P(x)-Q(y)=0, where P,Q are rational functions. We also apply the results obtained to the functional equations P(f)=Q(g) and P(f)=cP(g), where c\in C. For example, we show that for a generic pair of rational functions P,Q the first equation has no non-constant solutions f,g meromorphic on C whenever (°P-1)(°Q-1) \geq 2.

math.CV↗
arXiv2008

On trees covering chains or stars

In this paper, in the context of the ``Dessins d'enfants'' theory, we give a combinatorial criterion for a plane tree to cover a tree from the classes of "chains" or "stars''. Besides, we discuss some applications of this result which are related to the arithmetical theory of torsion on curves.

math.AG↗