arXiv subjects
Publications and source records attributed to Fedor Pakovich.
We show that a variety of known and new results concerning connected plane graphs whose vertex and face degrees satisfy prescribed uniformity conditions with at most two exceptions can be deduced from recent results on the Hurwitz existence problem regarding the realizability of branch patterns of rational functions. Our method also yields a description of the Belyi functions corresponding to such graphs.
Let $X$ and $Y$ be rational functions of degree at least two with complex coefficients such that $\mathbb{C}(X,Y)=\mathbb{C}(z)$. We study the problem of determining when the field extension $[\mathbb{C}(z):\mathbb{C}(X)\cap\mathbb{C}(Y)]$ is finite and attains the minimal possible degree ${\rm deg X}\cdot{\rm deg Y}$. We give a complete characterization in the case where $X$ is a Galois covering. We also establish several related results concerning the functional equation $A \circ X = Y \circ B$ in rational functions, in the case where one of the functions involved is a Galois covering. Finally, we consider an analogous problem for holomorphic maps between compact Riemann surfaces.
Let $ R$ be a compact Riemann surface, and let $ P: R \to \mathbb P^1(\mathbb C) $ and $ Q: R \to \mathbb P^1(\mathbb C) $ be holomorphic maps. In this paper, we investigate the following problem: under what conditions do the preimages $ P^{-1}(K) $ and $ Q^{-1}(K) $ coincide for some infinite set $K$ contained in $\mathbb P^1(k)$, where $k$ is a finitely generated subfield of $\mathbb C$ (e.g., a number field)? Equivalently, we study holomorphic correspondences that admit an infinite completely invariant set contained in $\mathbb P^1(k)$. We show that if such a set exists, then there is a holomorphic Galois covering $Θ: R_0 \to \mathbb P^1(\mathbb C)$, where $R_0$ has genus zero or one, such that $ P $ and $ Q $ are ``compositional left factors" of $ Θ.$ We also consider a more general equation $ P^{-1}(K_1) = Q^{-1}(K_2),$ where $K_1$ and $K_2$ are infinite subsets of $\mathbb P^1(k)$.
Let $A_1$ and $A_2$ be polynomials of degree at least two over $\mathbb C$. We say that $A_1$ and $A_2$ are intertwined if the endomorphism $(A_1, A_2)$ of $\mathbb C\mathbb P^1 \times \mathbb C\mathbb P^1$ given by $(z_1, z_2) \mapsto (A_1(z_1), A_2(z_2))$ admits an irreducible periodic curve that is neither a vertical nor a horizontal line. We denote by $\mathrm{Inter}(A)$ the set of all polynomials $B$ such that some iterate of $B$ is intertwined with some iterate of $A$. In this paper, we prove a conjecture of Favre and Gauthier describing the structure of $\mathrm{Inter}(A)$. We also obtain a bound on the possible periods of periodic curves for endomorphisms $(A_1, A_2)$ in terms of the sizes of the symmetry groups of the Julia sets of $A_1$ and $A_2$.
We describe algebraic curves $ X : F(x, y) = 0 $ defined over $\overline{\mathbb{Q}}$ that satisfy the following property: there exist a number field $k$ and an infinite set $S \subset k$ such that, for every $y \in S$, the roots of the polynomial $F(x, y)$ belong to $k$.
We show that for a general rational function $A$ of degree $m \geq 2$, any decomposition of its iterate $A^{\circ n}$, $n \geq 1$, into a composition of indecomposable rational functions is equivalent to the decomposition $A^{\circ n}$ itself. As an application, we prove that if $(A_1, A_2)$ is a pair of general rational functions, then the endomorphism of $\mathbb C\mathbb P^1 \times \mathbb C\mathbb P^1$ given by $ (z_1, z_2) \mapsto (A_1(z_1), A_2(z_2)) $ admits a periodic curve that is neither a vertical nor a horizontal line if and only if $A_1$ and $A_2$ are conjugate.
For a Belyi function $β:\mathbb C\mathbb P^1\rightarrow \mathbb C\mathbb P^1$ ramified only over the points $-1,1,\infty$, a corresponding ``dessin d'enfant'' $\mathcal D_β$ is defined as the set $β^{-1}([-1,1])$ considered as a bi-colored graph on the Riemann sphere whose white and black vertices are points of the sets $β^{-1}\{-1\}$ and $β^{-1}\{1\}$ correspondingly. Merely the set $β^{-1}([-1,1])$ without a graph structure is called a support of $\mathcal D_β$. In this note, we solve the following problem: under what conditions different dessins $\mathcal D_{β_1}$ and $\mathcal D_{β_2}$ have equal supports?
We give lower bounds for genera of components of fiber products of holomorphic maps between compact Riemann surfaces, extending results on genera of components of algebraic curves of the form $A(x)-B(y)=0,$ where $A$ and $B$ are rational functions.
Let $k$ be an algebraically closed field of characteristic zero, and $k[[z]]$ the ring of formal power series over $k$. In this paper, we study equations in the semigroup $z^2k[[z]]$ with the semigroup operation being composition. We prove a number of general results about such equations and provide some applications. In particular, we answer a question of Horwitz and Rubel about decompositions of ``even'' formal power series. We also show that every right amenable subsemigroup of $z^2k[[z]]$ is conjugate to a subsemigroup of the semigroup of monomials.
With each holomorphic map $f: R \rightarrow \mathbb C\mathbb P^1$, where $R$ is a compact Riemann surface, one can associate a combinatorial datum consisting of the genus $g$ of $R$, the degree $n$ of $f$, the number $q$ of branching points of $f$, and the $q$ partitions of $n$ given by the local degrees of $f$ at the preimages of the branching points. These quantities are related by the Riemann-Hurwitz formula, and the Hurwitz existence problem asks whether a combinatorial datum that fits this formula actually corresponds to some map $f$. In this paper, using results and techniques related to fiber products of holomorphic maps between compact Riemann surfaces, we prove a number of results that enable us to uniformly explain the non-realizability of many previously known non-realizable branch data, and to construct a large amount of new such data. We also deduce from our results the theorem of Halphen, proven in 1880, concerning polynomial solutions of the equation $A(z)^a+B(z)^b=C(z)^c$, where $a,b,c$ are integers greater than one.
Let $A$ and $B$ be non-constant rational functions over $\mathbb{C}$, and let $K \subset \mathbb{P}^1(\mathbb{C})$ be an infinite set. Using height functions, we prove that the inclusion $ A^{-1}(K) \subseteq B^{-1}(K) $ implies the inequality $ {\rm deg} B \geq {\rm deg} A $ in the following two cases: the set $K$ is contained in $\mathbb{P}^1(k)$, where $ k$ is a finitely generated subfield of $\mathbb{C}$, or the set $K$ is discrete in $\mathbb{C}$, and $A$ and $B$ are polynomials. In particular, this implies that for $A$, $B$, and $K$ as above, the equality $ A^{-1}(K) = B^{-1}(K) $ is impossible, unless $ {\rm deg} B = {\rm deg} A $.
Let $A$ be a rational function of one complex variable, and $z_0$ its repelling fixed point with the multiplier $λ.$ Then a Poincaré function associated with $z_0$ is a function $\mathcal{P}_{A,z_0,λ}$ meromorphic on $\mathbb C$ such that $\mathcal{P}_{A,z_0,λ}(0)=z_0$, $\mathcal{P}_{A,z_0,λ}'(0)\neq 0,$ and $\mathcal{P}_{A,z_0,λ}(λz)=A\circ \mathcal{P}_{A,z_0,λ}(z).$ In this paper, we investigate the following problem: given Poincaré functions $\mathcal{P}_{A_1,z_1,λ_1}$ and $\mathcal{P}_{A_2,z_2,λ_2}$, find out if there is an algebraic relation $f(\mathcal{P}_{A_1,z_1,λ_1},\mathcal{P}_{A_2,z_2,λ_2})=0$ between them and, if such a relation exists, describe the corresponding algebraic curve. We provide a solution, which can be viewed as a refinement of the classical theorem of Ritt about commuting rational functions. We also reprove and extend previous results concerning algebraic dependencies between Böttcher functions.
For a non-constant complex rational function $P$, the lemniscate of $P$ is defined as the set of points $z\in \mathbb C$ such that $\vert P(z)\vert =1$. The lemniscate of $P$ coincides with the set of real points of the algebraic curve given by the equation $L_P(x,y)=0$, where $L_P(x,y)$ is the numerator of the rational function $P(x+iy)\overline{ P}(x-iy)-1.$ In this paper, we study the following two questions: under what conditions two lemniscates have a common component, and under what conditions the algebraic curve $L_P(x,y)=0$ is irreducible. In particular, we provide a sharp bound for the number of complex solutions of the system $\vert P_1(z)\vert =\vert P_2(z)\vert =1$, where $P_1$ and $P_2$ are rational functions.
Let $F$ be a rational function of one complex variable of degree $m\geq 2$. The function $F$ is called simple if for every $z\in \mathbb C\mathbb P^1$ the preimage $F^{-1}\{z\}$ contains at least $m-1$ points. We show that if $F$ is a simple rational function of degree $m\geq 4$ and $F^{\circ l} =G_r\circ G_{r-1}\circ \dots \circ G_1$, $l\geq 1$, is a decomposition of an iterate of $F$ into a composition of indecomposable rational functions, then $r=l$ and there exist Möbius transformations $μ_i,$ $1\leq i \leq r-1,$ such that $G_r=F\circ μ_{r-1},$ $G_i=μ_{i}^{-1}\circ F \circ μ_{i-1},$ $1<i< r,$ and $G_1=μ_{1}^{-1}\circ F$. As applications, we solve a number of problems in complex and arithmetic dynamics for "general" rational functions.
Let $A$ be a rational function of degree $n\geq 2$. Let us denote by $ G(A)$ the group of Möbius transformations $σ$ such that $ A\circ σ=ν_σ \circ A$ for some Möbius transformations $ν_σ$, and by $Σ(A)$ and ${\rm Aut}(A)$ the subgroups of $ G(A)$ consisting of $σ$ such that $ A\circ σ= A$ and $ A\circ σ= σ\circ A$, correspondingly. In this paper, we study sequences of the above groups arising from iterating $A$. In particular, we show that if $A$ is not conjugate to $z^{\pm n},$ then the orders of the groups $ G(A^{\circ k})$, $k\geq 2,$ are finite and uniformly bounded in terms of $n$ only. We also prove a number of results about the groups $Σ_{\infty}(A)=\cup_{k=1}^{\infty} Σ(A^{\circ k})$ and ${\rm Aut}_{\infty}(A)=\cup_{k=1}^{\infty} {\rm Aut}(A^{\circ k})$, which are especially interesting from the dynamical perspective.
The concepts of differentiation and integration for matrices are known. As far as each matrix is differentiable, it is not clear a priori whether a given matrix is integrable or not. Recently some progress was obtained for diagonalizable matrices, however general problem remained open. In this paper, we present a full solution of the integrability problem. Namely, we provide necessary and sufficient conditions for a given matrix to be integrable in terms of its characteristic polynomial. Furthermore, we find necessary and sufficient conditions for the existence of integrable and non-integrable matrices with given geometric multiplicities of eigenvalues. Our approach relies on properties of some special classes of polynomials, namely, Shabat polynomials and conservative polynomials, arising in number theory and dynamics.
Let $k$ be an algebraically closed field of characteristic zero, and $k[[z]]$ the ring of formal power series over $k$. We provide several characterizations of right amenable finitely generated subsemigroups of $z^2k[[z]]$ with the semigroup operation $\circ $ being composition. In particular, we show that a subsemigroup $S=\langle Q_1,Q_2,\dots, Q_k\rangle$ of $z^2k[[z]]$ is right amenable if and only if there exists an invertible element $β$ of $zk[[z]]$ such that $β^{-1}\circ Q_i \circ β=ω_i z^{d_i},$ $1\leq i \leq k,$ for some integers $d_i$, $1\leq i \leq k,$ and roots of unity $ω_i,$ $1\leq i \leq k.$
Let $A$ be a rational function of degree at least two on the Riemann sphere. We say that $A$ is tame if the algebraic curve $A(x)-A(y)=0$ has no factors of genus zero or one distinct from the diagonal. In this paper, we show that if tame rational functions $A$ and $B$ have orbits with infinite intersection, then $A$ and $B$ have a common iterate. We also show that for a tame rational function $A$ decompositions of its iterates $A^{\circ d},$ $d\geq 1,$ into compositions of rational functions can be obtained from decompositions of a single iterate $A^{\circ N}$ for $N$ big enough.