340 problems

Finite-difference stability conjecture. For any stable polynomial $p$, the polynomials

Determine whether, for every countable family of hyperplanes $\{H_j\}_{j\in\mathbb{N}}$ in general position in $\mathbb{P}^m$ and every linearly nondegenerate holomorphic curve…

For every non-hyperbolic Riemann surface $X$ and every singular hyperbolic metric $g$ on $X$ in the sense of potential theory, the associated monodromy group…

For every finitely generated polynomial submodule $M\subset H^2(\mathbb D^2)$, Yang's numerical invariants satisfy $\Sigma_0(M)\geq \Sigma_1(M)\geq \Sigma_2(M)\geq\cdots$.

Let $f^\beta(x)$ be the unique solution of equation (2) with $f^\beta(0)=1$. Asymptotic growth conjecture. As $x\to+\infty$, the function $f^\beta$ should be asymptotic to … for a…

Let $f(z)=p(z)\overline{q(z)}$ be a logharmonic polynomial, where $p$ and $q$ are analytic polynomials of degrees $n$ and $m$, respectively, with $p$ not a constant multiple of $q$…

Khabibullin's conjecture. The stated integral implication holds for every such $S$, every $\rho>1$, and every integer $n\geq 2$. This is an extremal integral inequality arising in…

Let $X$ be the space and let $\vphi_L$, $\psi$, $\ell$, $\spH$, $\spE$, $\sigma$, $\sigma_{\operatorname{mlc}}$, $\lcc$, $\multidl_{}\paren{\vphi_L}$, and $\RTF|\cdot|(t)[\sigma]$…

Random Runge approximation conjecture. There is a sequence $R_k(\omega,z)$ of random rational functions such that, for each $\omega$, $R_k(\omega,\cdot)\rightarrow f(\omega,\cdot)$…

Converse characterization conjecture. If $f$ is an extreme point of $PRP_2$, then $p$ is $\mathbb{T}^2$-saturated and $p-\widetilde{p}$ is irreducible.

Let $f$ be holomorphic on a neighborhood of the closure of the unit disk $\mathbb{D}$. An injective holomorphic function is a holomorphic map that is one-to-one. Conformal polynomi…

Uniqueness conjecture. If

Let $z=(z_1,z_2,\dots,z_n)$ be free commutative variables, let $D_i=\frac{\partial}{\partial z_i}$, and let … be the Laplace operator on $\mathbb C[z]$. A polynomial $P(z)$ is Hess…

Let $\Omega\subset\subset D$ be convex domains with $C^2$ boundaries in $\mathbb{C}^n$, and let $f$ be continuous on $bD$. Globevnik–Stout conjecture. If $f$ extends continuously t…

Equivalent half-plane formulation. The function $F$ has infinitely many zeros in $\operatorname{Re}w>\frac12$.

Borcea's conjecture. The function $f$ has infinitely many zeros in $|z|<\rho$.

Conjecture on infinitely many zeros. The function $f$ has infinitely many zeros in $\mathbb C$.

Let $S(n,0)$ be the set of monic complex polynomials of degree $n\ge 2$ whose zeros lie in the closed unit disk and which vanish at $0$. For $p\in S_n$, define … The paper has dete…

Let $\beta\in\bar{D}$ and let $S(n,\beta)$ be the set of monic complex polynomials in $S_n$ having at least one zero at $\beta$. For $\alpha\in\mathbf{C}$ and $p\in S(n,\beta)$, de…

Let $S_n$ be the set of monic complex polynomials of degree $n\ge 2$ whose zeros lie in the closed unit disk. For $p\in S_n$, define … Call $p$ extremal for Sendov's conjecture if…

Complex-charge equilibrium zero conjecture. The meromorphic function

Let $\{\alpha_n\}_{n\in\mathbb{N}}$ be unimodular complex numbers and let $\{\beta_n\}_{n\in\mathbb{N}}$ be positive numbers converging to $0$, with the ratios…

Let $\{z_n\}_{n\in\mathbb{N}}$ be distinct points in the open unit disk converging to $1$. Let $\mathcal{H}$ be a separable infinite-dimensional complex Hilbert space with orthonor…

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…

Generalized spiral limit conjecture. The rescaled functions $p_k\left(z_0+\frac{z}{\sqrt{k}}\right)$ have a subsequence converging to a normalization of $P_\lambda$ for some…