We consider the existence problem of meromorphic solutions of the Fermat-type difference equation

$$ \begin{align*} f(z)^p+f(z+c)^q=h(z), \end{align*} $$

where $p,q$ are positive integers, and h has few zeros and poles in the sense that $N(r,h) + N(r,1/h) = S(r,h)$. As a particular case, we consider $h=e^g$, where g is an entire function. Additionally, we briefly discuss the case where h is small with respect to f in the standard sense $T(r,h)=S(r,f)$.

This paper consists of three parts: First, letting $b_1(z)$, $b_2(z)$, $p_1(z)$ and $p_2(z)$ be nonzero polynomials such that $p_1(z)$ and $p_2(z)$ have the same degree $k\geq 1$ and distinct leading coefficients $1$ and $\alpha$, respectively, we solve entire solutions of the Tumura–Clunie type differential equation $f^{n}+P(z,\,f)=b_1(z)e^{p_1(z)}+b_2(z)e^{p_2(z)}$, where $n\geq 2$ is an integer, $P(z,\,f)$ is a differential polynomial in $f$ of degree $\leq n-1$ with coefficients having polynomial growth. Second, we study the oscillation of the second-order differential equation $f''-[b_1(z)e^{p_1(z)}+b_2(z)e^{p_2(z)}]f=0$ and prove that $\alpha =[2(m+1)-1]/[2(m+1)]$ for some integer $m\geq 0$ if this equation admits a nontrivial solution such that $\lambda (f)<\infty$. This partially answers a question of Ishizaki. Finally, letting $b_2\not =0$ and $b_3$ be constants and $l$ and $s$ be relatively prime integers such that $l> s\geq 1$, we prove that $l=2$ if the equation $f''-(e^{lz}+b_2e^{sz}+b_3)f=0$ admits two linearly independent solutions $f_1$ and $f_2$ such that $\max \{\lambda (f_1),\,\lambda (f_2)\}<\infty$. In particular, we precisely characterize all solutions such that $\lambda (f)<\infty$ when $l=2$ and $l=4$.

We investigate Carlson–Griffiths’ equidistribution theory of meormorphic mappings from a complete Kähler manifold into a complex projective algebraic manifold. By using a technique of Brownian motions developed by Atsuji, we obtain a second main theorem in Nevanlinna theory provided that the source manifold is of nonpositive sectional curvature. In particular, a defect relation follows if some growth condition is imposed.

We prove several results on unavoidable families of meromorphic functions. For instance, we give new examples of families of cardinality 3 that are unavoidable with respect to the set of meromorphic functions on $\mathbb C$ . We further obtain families consisting of less than three functions that are unavoidable with respect to certain subsets of meromorphic functions. In the other direction, we show that for every meromorphic function f, there exists an entire function that avoids f on $\mathbb C$ .

We construct a family of fibred threefolds $X_m \to (S , \Delta )$ such that $X_m$ has no étale cover that dominates a variety of general type but it dominates the orbifold $(S,\Delta )$ of general type. Following Campana, the threefolds $X_m$ are called weakly special but not special. The Weak Specialness Conjecture predicts that a weakly special variety defined over a number field has a potentially dense set of rational points. We prove that if m is big enough, the threefolds $X_m$ present behaviours that contradict the function field and analytic analogue of the Weak Specialness Conjecture. We prove our results by adapting the recent method of Ru and Vojta. We also formulate some generalisations of known conjectures on exceptional loci that fit into Campana’s program and prove some cases over function fields.

The Fermat type functional equations $(*)\, f_1^n+f_2^n+\cdots +f_k^n=1$, where n and k are positive integers, are considered in the complex plane. Our focus is on equations of the form (*) where it is not known whether there exist non-constant solutions in one or more of the following four classes of functions: meromorphic functions, rational functions, entire functions, polynomials. For such equations, we obtain estimates on Nevanlinna functions that transcendental solutions of (*) would have to satisfy, as well as analogous estimates for non-constant rational solutions. As an application, it is shown that transcendental entire solutions of (*) when n = k(k − 1) with k ≥ 3, would have to satisfy a certain differential equation, which is a generalization of the known result when k = 3. Alternative proofs for the known non-existence theorems for entire and polynomial solutions of (*) are given. Moreover, some restrictions on degrees of polynomial solutions are discussed.

The Brück conjecture states that if a nonconstant entire function $f$ with hyper-order ${\it\sigma}_{2}(f)\in [0,+\infty )\setminus \mathbb{N}$ shares one finite value $a$ (counting multiplicities) with its derivative $f^{\prime }$, then $f^{\prime }-a=c(f-a)$, where $c$ is a nonzero constant. The conjecture has been established for entire functions with order ${\it\sigma}(f)<+\infty$ and hyper-order ${\it\sigma}_{2}(f)<{\textstyle \frac{1}{2}}$. The purpose of this paper is to prove the Brück conjecture for the case ${\it\sigma}_{2}(f)=\frac{1}{2}$ by studying the infinite hyper-order solutions of the linear differential equations $f^{(k)}+A(z)f=Q(z)$. The shared value $a$ is extended to be a ‘small’ function with respect to the entire function $f$.

In this note, we prove a uniqueness theorem for finite-order meromorphic solutions to a class of difference equations of Malmquist type. Such solutions $f$ are uniquely determined by their poles and the zeros of $f-e_{j}$ (counting multiplicities) for two finite complex numbers $e_{1}\neq e_{2}$.

We consider solutions to the algebraic differential equation $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}f^nf'+Q_d(z,f)=u(z)e^{v(z)}$, where $Q_d(z,f)$ is a differential polynomial in $f$ of degree $d$ with rational function coefficients, $u$ is a nonzero rational function and $v$ is a nonconstant polynomial. In this paper, we prove that if $n\ge d+1$ and if it admits a meromorphic solution $f$ with finitely many poles, then

$$\begin{equation*} f(z)=s(z)e^{v(z)/(n+1)} \quad \mbox {and}\quad Q_d(z,f)\equiv 0. \end{equation*}$$

$f$

$f'p_k(f)+q_m(f)$

$\alpha $

$p_k(f), q_m(f)$

$f$

$k$

$m$

$k\ge m+1$

It is a well-known result that if a nonconstant meromorphic function $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}f$ on $\mathbb{C}$ and its $l$th derivative $f^{(l)}$ have no zeros for some $l\geq 2$, then $f$ is of the form $f(z)=\exp (Az+B)$ or $f(z)=(Az+B)^{-n}$ for some constants $A$, $B$. We extend this result to meromorphic functions of several variables, by first extending the classic Tumura–Clunie theorem for meromorphic functions of one complex variable to that of meromorphic functions of several complex variables using Nevanlinna theory.

Working from a half-plane result of Fletcher and Langley, we show that if f is an integer-valued function on some subset of the natural numbers of positive lower density and is meromorphic of sufficiently small exponential type in the plane, then f is a polynomial.

Let $f$ be meromorphic of finite order in the plane, such that $f^{(k)}$ has finitely many zeros, for some $k\geq2$. The author has conjectured that $f$ then has finitely many poles. In this paper, we strengthen a previous estimate for the frequency of distinct poles of $f$. Further, we show that the conjecture is true if either

1. $f$ has order less than $1+\varepsilon$, for some positive absolute constant $\varepsilon$, or

2. $f^{(m)}$, for some $0\leq m lt k$, has few zeros away from the real axis.

AMS 2000 Mathematics subject classification: Primary 30D35

A result is presented giving conditions on a set of open discs in the complex plane that ensure that a transcendental meromorphic function with Nevanlinna deficient poles omits at most one finite value outside the set of discs. This improves a previous result of Langley, and goes some way towards closing a gap between Langley's result and a theorem of Toppila in which the omitted values considered may include ∞

In this note, we give a new short proof of the fact, recently discovered by Ye, that all (finite) values are equidistributed by the Riemann zeta function.

A complex projective manifold is said to be weakly special if none of its finite étale covers has a dominant rational map to a positive-dimensional manifold of general type. Weakly special manifolds are conjectured by Harris and Tschinkel (Rational points on quartics, Duke Math. J. 104 (2000), 477–500) to be potentially dense if defined over a number field. In a previous paper by F. Campana (Special varieties, orbifolds and classification theory, Ann. Inst. Fourier (Grenoble) 54 (2004), 499–665), the stronger notion of special (complex projective) manifolds was introduced. These are conjectured in (9.20) of that work to be exactly the potentially dense manifolds (when defined over a number field). The two notions of specialness and weak-specialness coincide up to dimension 2, but differ from dimension 3 on, as shown by examples $X$ constructed by Bogomolov and Tschinkel (Special elliptic fibrations, in Proc. Fanu Conf., Torino, 2003, ed. A. Conte), answering a question raised in Campana's paper. These examples should thus be potentially dense, according to the first conjecture, but should not be so, according to the second conjecture. The present techniques of arithmetic geometry do not seem to be able to decide between these two conflicting conjectures. So, instead of this, we investigate the more tractable hyperbolic aspects of the examples of Bogomolov and Tschinkel, and show that for certain manifolds $X$, the behaviour of entire transcendental curves is that given in Conjecture 9.16, p. 618 of Campana's above-mentioned work. More precisely, the image of any holomorphic map from ${\mathbb{C}}$ to $X$ is contained either in some fibre of the unique elliptic fibration on $X$, or in some fixed divisor of $X$. This is consistent with the standard links (Lang and Vojta for varieties of general type, the first author's previous work in general) expected between the hyperbolic and arithmetic behaviour of projective varieties in cases where the second conjecture (stated by Campana) holds. Conversely, if the conjecture stated by Harris and Tschinkel is true, these expected links between arithmetics and complex hyperbolicity should fail to hold on these examples already.

In this paper, Hinkkanen's problem (1984) is completely solved, i.e., it is shown that any meromorphic function $f$ is determined by its zeros and poles and the zeros of ${{f}^{\left( j \right)}}$ for $j=1,2,3,4$.

In this paper we investigate the uniqueness of transcendental meromorphic function dealing with the shared values in some angular domains instead of the whole complex plane.

In this paper we treat two non-linear differential equations which come from complex dynamics theory. We give a complete classification of the equations when they possess transcendental meromorphic solutions.