Let K be a non archimedean algebraically closed field of characteristic pi complete for its ultrametric absolute value. In a recent paper by Escassut and Yang, polynomial decompositions P(f)=Q(g) for meromorphic functions f, g on K (resp. in a disk) have been considered, and for a class of polynomials P, Q, estimates for the Nevanlinna function T(r,f) have been derived. In the present paper we consider as a generalization rational decompositions of meromorphic functions. In the case, where f, g are analytic functions, the Second Nevanlinna Theorem yields an analogue result as in the mentioned paper. However, if they are meromorphic, non trivial estimates for T(r,f) are more sophisticated.
In this paper, we study primeness and pseudo primeness of p-adic meromorphic functions. We also consider left (resp. right ) primeness of these functions. We give, in particular, sufficient conditions for a meromorphic function to satisfy such properties. Finally, we consider the problem of permutability of entire functions.
We prove a result on the representation of squares by second degree polynomials in the field of $p$-adic meromorphic functions in order to solve positively Büchi's $n$ squares problem in this field (that is, the problem of the existence of a constant $M$ such that any sequence $(x_n^2)$ of $M$ - not all constant - squares whose second difference is the constant sequence $(2)$ satisfies $x_n^2=(x+n)^2$ for some $x$). We prove (based on works by Vojta) an analogous result for function fields of characteristic zero, and under a Conjecture by Bombieri, an analogous result for number fields. Using …
Let h be a complex meromorphic function decomposed in two different ways P(f) and Q(g), where f, g are meromorphic functions and P, Q are rational functions. We follow an approach due to C.-C. Yang, P. Li and K. H. Ha who handle similar decompositions, however, with P, Q polynomials, by applying the Second Nevanlinna Theorem. We establish relations between the number k of distinct zeros of P', the degrees of P and Q, and (unless f, g are analytic functions) the number of distinct zeros of the denominators of P and Q.
We find all non-rational meromorphic solutions of the equation $ww"-(w')^2=α(z)w+β(z)w'+γ(z)$, where $α$, $β$ and $γ$ are rational functions of $z$. In so doing we answer a question of Hayman by showing that all such solutions have finite order. Apart from special choices of the coefficient functions, the general solution is not meromorphic and contains movable branch points. For some choices for the coefficient functions the equation admits a one-parameter family of non-rational meromorphic solutions. Nevanlinna theory is used to show that all such solutions have been found and allows us to a…
We show that an $n$-dimensional compact Kähler manifold $X$ admitting a non-degenerate meromorphic map $f:{\bf C}^n\to X$ of order $ρ_f<2$ is rationally connected.
It is shown that if It is shown that if \begin{equation}\label{abstract_eq} f(z+1)^n=R(z,f),\tag† \end{equation} where $R(z,f)$ is rational in $f$ with meromorphic coefficients and $°_f(R(z,f))=n$, has an admissible meromorphic solution, then either $f$ satisfies a difference linear or Riccati equation with meromorphic coefficients, or \eqref{abstract_eq} can be transformed into one in a list of ten equations with certain meromorphic or algebroid coefficients. In particular, if \eqref{abstract_eq}, where the assumption $°_f(R(z,f))=n$ has been discarded, has rational coefficients and a transce…
We study one variable meromorphic functions mapping a planar real algebraic set $A$ to another real algebraic set in the complex plane. By using the theory of Schwarz reflection functions, we show that for certain $A$, these meromorphic functions must be rational. In particular, when $A$ is the standard unit circle, we obtain an one dimensional analog of Poincaré(1907), Tanaka(1962) and Alexander(1974)'s rationality results for $2m-1$ dimensional sphere in $\mathbb{C}^m$ when $m\ge 2$.
We study the convergence of sequences of type I and type II Hermite-Padé approximants for certain systems of meromorphic functions made up of rational modifications of Nikishin systems of functions.
If the preimage of a four-point set under a meromorphic function belongs to the real line, then the image of the real line is contained in a circle in the Riemann sphere. We include an application of this result to holomorphic dynamics: if the Julia set of a rational function is contained in a smooth curve then it is contained in a circle.
The paper determines all meromorphic functions with finitely many zeros in the plane having the property that a linear differential polynomial in the function, of order at least 3 and with rational functions as coefficients, also has finitely many zeros.
We study the convergence of type I Hermite-Padé approximation for a class of meromorphic functions obtained by adding a vector of rational functions with real coefficients to a Nikishin system of functions.
Necessary conditions are obtained for certain types of rational delay differential equations to admit a non-rational meromorphic solution of hyper-order less than one. The equations obtained include delay Painlevé equations and equations solved by elliptic functions.
The present paper deals with the convergence properties of multi-level Hermite-Padé approximants for a class of meromorphic functions given by rational perturbations with real coefficients of a Nikishin system of functions, and study the zero location of the corresponding multiple orthogonal polynomials.
Meromorphic solutions of non-linear differential equations of the form $f^n+P(z,f)=h$ are investigated, where $n\geq 2$ is an integer, $h$ is a meromorphic function, and $P(z,f)$ is differential polynomial in $f$ and its derivatives with small functions as its coefficients. In the existing literature this equation has been studied in the case when $h$ has the particular form $h(z)=p_1(z)e^{α_1(z)}+p_2(z)e^{α_2(z)}$, where $p_1, p_2$ are small functions of $f$ and $α_1,α_2$ are entire functions. In such a case the order of $h$ is either a positive integer or equal to infinity. In this article i…
Let $ι: \C^2 \hookrightarrow S$ be a compactification of the two dimensional complex space $\C^2$. By making use of Nevanlinna theoretic methods and the classification of compact complex surfaces K. Kodaira proved in 1971 (\cite{ko71}) that $S$ is a rational surface. Here we deal with a more general meromorphic map $f: \C^n \to X$ into a compact complex manifold $X$ of dimension $n$, whose differential $df$ has generically rank $n$. Let $ρ_f$ denote the order of $f$. We will prove that if $ρ_f<2$, then every global symmetric holomorphic tensor must vanish; in particular, {\it if $\dim X=2$ and…
In this paper, we study the transcendental meromorphic solutions for the nonlinear differential equations: $f^{n}+P(f)=R(z)e^{α(z)}$ and $f^{n}+P_{*}(f)=p_{1}(z)e^{α_{1}(z)}+p_{2}(z)e^{α_{2}(z)}$ in the complex plane, where $P(f)$ and $P_{*}(f)$ are differential polynomials in $f$ of degree $n-1$ with coefficients being small functions and rational functions respectively, $R$ is a non-vanishing small function of $f$, $α$ is a nonconstant entire function, $p_{1}, p_{2}$ are non-vanishing rational functions, and $α_{1}, α_{2}$ are nonconstant polynomials. Particularly, we consider the solutions …
We study the class $\mathcal{M}$ of functions meromorphic outside a countable closed set of essential singularities. We show that if a function in $\mathcal{M}$, with at least one essential singularity, permutes with a non-constant rational map $g$, then $g$ is a Möbius map that is not conjugate to an irrational rotation. For a given function $ f \in\mathcal{M}$ which is not a Möbius map, we show that the set of functions in $\mathcal{M}$ that permute with $ f $ is countably infinite. Finally, we show that there exist transcendental meromorphic functions $f: \mathbb{C} \to \mathbb{C}$ such tha…
This paper attempts to describe some of the results obtained in the iteration theory of transcendental meromorphic functions, not excluding the case of entire functions. The reader is not expected to be familiar with the iteration theory of rational functions. On the other hand, some aspects where the transcendental case is analogous to the rational case are treated rather briefly here. For example, we introduce the different types of components of the Fatou set that occur in the iteration of rational functions but omit a detailed description of these types. Instead, we concentrate on the type…
We construct some explicit formulas of rational maps and transcendental meromorphic functions having Herman rings of period strictly larger than one. This gives an answer to a question raised by Shishikura in the 1980s. Moreover, the formulas of some rational maps with nested Herman rings are also found. To obtain the formulas of transcendental meromorphic functions having periodic Herman rings, a crucial step is to find an explicit family of transcendental entire functions having bounded Siegel disks of all possible periods and rotation numbers. This is based on proving the existence of a Man…
A proper or singular abelian mapping from $C^n$ to $\bar\C^n$ is parametrized by $n$ meromorphic functions with at most $2n$ periods. We develop the existence and structure theorems of the classical theory of an abelian mapping purely on the basis of its defining functional equation, the so-called algebraic addition theorem (AAT), with no appeal to any representation as quotients of theta functions. We offer two new proofs of the periodicity of a nonrational mapping admitting an AAT. We also prove by new arguments the existence of a rational group law on an associated algebraic variety, and th…
We introduce a general third order non-linear autonomous ODE which covers many ODEs coming from boundary layer problems, like the Falkner-Skan equation and the Cheng-Minkowycz equation. Using Wiman-Valiron theory and complex analytic methods recently developed, for the generic cases, it is shown that all their meromorphic solutions must be rational, or rational in one exponential, and then we find all of them explicitly. For a few non-generic cases, some solutions, which are meromorphic or singlevalued, are also obtained. Our results also explain why it is so difficult to obtain new closed-for…
This paper is part of a program to understand the parameter spaces of dynamical systems generated by meromorphic functions with finitely many singular values. We give a full description of the parameter space for a specific family based on the exponential function that has precisely two finite asymptotic values and one attracting fixed point. It represents a step beyond the previous work in [GK] on degree 2 rational functions with analogous constraints: two critical values and an attracting fixed point. What is interesting and promising for pushing the general program even further, is that, de…
Sufficient boundary asymptotic conditions are established for a generalized Schur function $f$ to be identically equal to a given rational function $g$ unimodular on the unit circle. Similar rigidity statements are presented for generalized Carathéodory and generalized Nevanlinna functions.
After introducing q-analogues of the Borel and Laplace transformations, we prove that to every formal power series solution of a linear q-difference equation with rational coefficients, we may apply several q-Borel and Laplace transformations of convenient orders and convenient direction in order to construct a solution of the same equation that is meromorphic on $\mathbb{C}^{*}$. We use this theorem to construct explicitly an invertible matrix solution of a linear q-difference system with rational coefficients, of which entries are meromorphic on $\mathbb{C}^{*}$. Moreover, when the system is…
Meromorphic solutions of first-order differential equations with rational exponential coefficients
Institute of Applied Mathematics, Shenzhen Polytechnic University（深圳职业技术大学应用数学研究所）；School of Mathematics and Information Science, Guangzhou University（广州大学数学与信息科学学院）；School of Basic and General Education, Guangzhou University of Software（广州软件学院基础与通识教育学院）
We show that any dynamics on any discrete planar sequence $S$ can be realized by the postsingular dynamics of some transcendental meromorphic function, provided we allow for small perturbations of $S$. This work was influenced by an analogous result of DeMarco, Koch and McMullen for finite $S$ in the rational setting. The proof contains a method for constructing meromorphic functions with good control over both the postsingular set of $f$ and the geometry of $f$, using the Folding Theorem of Bishop and a classical fixpoint theorem of Tychonoff.
We use contact geometry to describe the monoid of projectively equivariant meromorphic differential operators on a complex curve, quantization of which generalizes known constructions of classical equivariants to non-commutative function algebras in several variables.
Let $M$ be a smooth compact $CR$ manifold of $CR$ dimension $n$ and $CR$ codimension $k$, which has a certain local extension property $E$. In particular, if $M$ is pseudoconcave, it has property $E$. Then the field $\Cal K(M)$ of $CR$ meromorphic functions on $M$ has transcendence degree $d$, with $d\leq n+k$. If $f_1, f_2, \hdots , f_d$ is a maximal set of algebraically independent $CR$ meromorphic functions on $M$, then $\Cal K(M)$ is a simple finite algebraic extension of the field $\Bbb C(f_1, f_2, \hdots, f_d)$ of rational functions of the $f_1, f_2, \hdots , f_d$. When $M$ has a project…
Let $X$ be a compact Kähler manifold and let $f:X\rightarrow X$ be a dominant rational map which is 1-stable. Let $λ_1$ and $λ_2$ be the first and second dynamical degrees of $f$. If $λ_1^2>λ_2$, then we show that $λ_1$ is a simple eigenvalue of $f^*:H^{1,1}(X)\rightarrow H^{1,1}(X)$, and moreover the unique eigenvalue of modulus $>\sqrt{λ_2}$. A variant of the result, where we consider the first spectral radius in the case the map $f$ may not be 1-stable, is also given. An application is stated for bimeromorphic selfmaps of 3-folds. In the last section of the paper, we prove analogs of the ab…
In this note, we give an elementary proof of the absence of invariant line fields on the conical Julia set of an analytic function of one variable. This proof applies not only to rational as well as transcendental meromorphic functions (where it was previously known), but even to the extremely general setting of Ahlfors islands maps as defined by Adam Epstein. In fact, we prove a more general result on the absence of invariant_differentials_, measurable with respect to a conformal measure that is supported on the (unbranched) conical Julia set. This includes the study of cohomological equation…
We prove the existence of (non compact) complex surfaces with a smooth rational curve embedded such that there does not exist any formal singular foliation along the curve. In particular, at arbitray small neighborhood of the curve, any meromorphic function is constant. This implies that the Picard group is not countably generated.
