19 problems

Complex-charge equilibrium zero conjecture. The meromorphic function

Let $\{a_i\}_{i\in\mathbb{N}}$ be positive with finite sum, and let $\{z_i\}_{i\in\mathbb{N}}$ be distinct complex numbers such that … for some $\rho>0$. Borel-series zero conjectu…

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…

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…

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.

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

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…

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…

Clunie–Eremenko–Rossi conjecture. The function $f(z)$ has infinitely many zeros.

Let $f$ be a real transcendental meromorphic function in the plane with at least one pole, meaning that $f(\mathbb R)\subseteq\mathbb R\cup\{\infty\}$. Assume that all zeros and po…

Hellerstein–Sheil-Small–Wittich conjecture. Then $f$ satisfies

Let $f$ be a general meromorphic function. An omitted point is a point $z_0\in\mathbb{C}$ such that $f(z)\neq z_0$ for every $z\in\mathbb{C}$. A Herman ring is a periodic doubly co…

Pole cancellation conjecture. The functions

Let $\Lambda$ be a dynamically natural slice for a family of meromorphic maps $\{f_\lambda,\ \lambda \in \Lambda\}$ in $\mathcal{M}_\infty$ in the strict sense. Let…

Hellerstein–Sheil-Small–Wiman conjecture. The function $f$ satisfies

Let $f=\frac{P}{Q}$ be a germ of a meromorphic function on $({\mathbb C}^{n+1},0)$, and let $Z_{\operatorname{top},f}$ denote its topological zeta function. Let $h_f$ be the monodr…

Let $\gamma$ be a pluripolar curve, let $\hat{\gamma}$ denote its projective hull, and let $\Sigma$ be a set contained in $\hat{\gamma}$. Let $\phi$ be the function associated with…