54 problems

Hayman's conjecture. The differential polynomial $f^n f'$ cannot have any non-zero generalized Picard exceptional values.

Consider the meromorphic Cauchy transform … where $|t_n|\to\infty$ and the series is understood under the usual local convergence condition. Clunie–Eremenko–Rossi conjecture. If…

Let $f$ be a meromorphic function on a disk in $\mathbb{C}$, and let $[p/q]_f$ denote its Padé approximant of order $(p,q)$. The diagonal approximants are $[p/p]_f$. Baker–Gammel–W…

Let $f$ be a transcendental entire function and let $k$ be a positive integer. Yang's periodicity conjecture. If $k\geq 1$ and $ff^{(k)}$ is periodic, then $f$ must be periodic. Th…

Let $f$ and $g$ be meromorphic functions on ${\mathbb{C}}$, and let $T(r,f)$ and $T(r,g)$ denote their Nevanlinna characteristic functions. Suppose that $f$ and $g$ share three dis…

Consider the simplest equation … obtained from Hayman's equation by setting $a_2=a_3=0$, where $a_1$, $b_1$, $b_2$, $b_3$, and $b_4$ are polynomials in $z$ and $a_1\not\equiv0$. Ha…

A generalized Briot–Bouquet equation is an equation … where $F$ is a polynomial in two variables and $k$ is an even or odd integer; $W$ denotes the class of functions under conside…

Naive-function-field conjecture. For arbitrary $Z$, one has

Complex-charge equilibrium zero conjecture. The meromorphic function

The setting is that of the paper's Ahlfors islands theorem, where $K$ is a non-archimedean field and $C_2$ is the lower-bound constant appearing in the hypothesis…

Let $N$ be a hyper-Kähler manifold of complex dimension $2n$, and suppose its Néron–Severi group has rank one and is generated by an isotropic element of $H^{2}(N,\mathbb{Z})$. The…

The second Painlevé hierarchy is an infinite sequence of nonlinear ordinary differential equations containing the second Painlevé equation … where $V=V(x)$ and $\alpha$ is constant…

Let $f$ be a transcendental entire function and let $k$ be a positive integer. For $k\geq 2$, suppose that either $ff^{(k)}$ is periodic or … for some entire function $h$. Yang's p…

Generic simplicity conjecture. We conjecture that the exceptional set $\mathcal{E}$ has measure zero in parameter space.

Let $X\subseteq\mathbb{C}$ be a connected open set and let $f:X\to\mathbb{C}$ be a meromorphic elementary function. A function is solvable when its equation with a constant right-h…

Yang–Li's conjecture. The equation has no transcendental meromorphic solution when $P_1/P_3$ is a non-zero constant and $P_2/P_3$ is not the square of any rational function.

Meromorphicity conjecture. For all $V\in\mathcal{C}$, not necessarily irreducible, and for generic quantum symmetric pair parameters, $K_V(z)$ is the Laurent series expansion of a…

The paper considers identities derived for periodic kernels and the corresponding identities for meromorphic kernels. Periodic-to-meromorphic kernel transfer conjecture. All identi…

Let $f$ be a meromorphic function, let $a\in\mathbf{C}$ satisfy $a\neq 1$ and $a\neq (n+1)/n$ for every $n\in\mathbf{N}$, and consider the assertion that $ff”-a(f')^2$ has infinite…

Let $f$ be a meromorphic function, and let $\overline{N}(r,\infty,f)$ denote the averaged counting function of distinct poles, counted without multiplicity. Goldberg's conjecture.…

Let $f$ and $g$ be two distinct non-constant meromorphic functions of infinite order, and let $a_1,a_2,a_3,a_4$ be four distinct values in the extended complex plane. Nonexistence…

Let $f$ and $g$ be two distinct transcendental meromorphic functions of infinite order, and let $a_1,a_2,a_3,a_4$ be four distinct values in the extended complex plane. Four-value…

Let $f$ be a meromorphic solution of the second-order differential equation … where $\Omega$ is polynomial in all of its arguments. Bank's conjecture. The characteristic function o…

Consider the class of differential equations … where $F$ is rational in its three arguments, and suppose the equation has a particular nonrational meromorphic solution. Nevanlinna'…

Let $f$ be meromorphic in the unit disk $\mathbb{D}=\{z:|z|<1\}$, have a pole at $z=p$, and have Taylor expansion … Suppose that, for some $0<\lambda<1$, … and denote this class by…