34 problems

Let $\mu$ be a probability measure on $\mathbb{C}$, let $X_1,X_2,\ldots$ be independent and identically distributed with law $\mu$, and define $P_n(z)=\prod_{j=1}^n(z-X_j)$. Let ……

Let $\{f_i\}_{i=1}^n$ be a system of functions satisfying condition (d) with parameter $p\in[0,1/2)$. For each fixed $x\in X$, let $f_k^*(x)$ denote the decreasingly ordered values…

Montgomery-Smith–Semenov hypothesis. There exist signs $\theta_i\in\{-1,1\}$ and a constant $c_0>0$ such that

Let $A_n$ and $B_n$ be independent GOE matrices, and write the normalized complex Gaussian pencil as $\widehat C_n(\lambda)=(A_n+\lambda B_n)/\sqrt{n(1+|\lambda|^2)}$. Its entry ps…

Let $A$ and $B$ be independently chosen from the real Gaussian orthogonal ensemble $GOE_n^{\mathbb R}$, consisting of real symmetric $n\times n$ matrices with the standard GOE dist…

Let $N_{\mathbb{R},\boldsymbol{\xi}}$ be the number of real zeros of a random polynomial from any model considered in Section. Assume that the coefficient distribution is subgaussi…

Let $P_n(x)=\sum_{k=0}^{n}\xi_kx^k$ be a Kac random polynomial, where the coefficients $\xi_k$ are independent and identically distributed copies of a random variable $\xi$. Assume…

Limiting root-measure conjecture. There is a compactly supported measure $\nu_\infty$ on $\mathbb{C}$ such that, for almost every sequence $(g_i)_{i\in\mathbb{N}}\in\Omega$,…

Concentration conjecture. Under mild conditions, the probability that the number of real zeros deviates from its expectation by at least $\varepsilon\log n$ is bounded above and be…

O. Nguyen's conjecture. Given $\alpha\in(0,1)$, there exists a bounded sequence $(c_k)$ of real numbers such that

Let $T_t:\mathbb{C}\to\mathbb{C}$ be the deterministic transport maps associated with the hydrodynamic limit, let $t_{\mathrm{sing}}$ be the first singular time, and let $z_j(t)$ d…

Let $\nu$ be a rotationally invariant probability measure on $\mathbb{C}\backslash\{0\}$ satisfying $\int_{0}^{\varepsilon} \nu(\{|z| < r\})r^{-1}\,\operatorname{d}r<\infty$ for so…

Let the coefficients of the harmonic polynomial model be independent and identically distributed random variables with mean zero, positive variance, and finite moments. The i.i.d.…

For $i=1,\ldots,n$, let $A_i\subseteq\mathbb{R}^n$ be finite nonempty sets of cardinality $t_i$, with convex hulls $P_i$. Let $V_0$ be the number of vertices of the Minkowski sum…

Let $p_d$ be a monic polynomial of degree $d$ with non-negative real roots … Let $\mu\in\mathcal{P}_{+,c}$, and suppose that the empirical root distributions of $p_d$ converge weak…

Fix $n\geq4$. Let $f$ be a degree-$n$ polynomial whose coefficients are chosen independently and uniformly from $[-L,L]\cap\mathbb{Z}$, and let $G_f$ denote its Galois group. The a…

Let $(\xi_j)_{j\geq 1}$ be independent complex-valued random variables with distribution $\mu$, let $P_n(z)=\prod_{j=1}^n(z-\xi_j)$, and let $\mu\lsem P_n^{(k)}\rsem$ denote the em…

Let $\x03be$ be a probability measure compactly supported in $\x1d53d$, let $P_n=\x220f_{i=1}^n(x-\x03be_i)$ have roots independently sampled according to $\x03be$, and let…

Let $P_n(z)=\prod_{k=1}^n(z-Z_k)$, where $Z_1,\ldots,Z_n$ are i.i.d. random variables with rotationally invariant law $\mu_0$ on $\mathbb{C}$ having no atom at $0$. Let $\mu_t$ be…

Let $\mathbb{T}_0$ be the domain of the ordered $k$-tuples used in the paper, and let $p_k(\mathbf{x},\mathbf{y})$ denote the corresponding joint density. Sharp density conjecture.…

Let $f_n(z)=\sum_{j=0}^n\varepsilon_j z^j$, where $\{\varepsilon_j\}_j$ are iid real random variables satisfying $\mathbb{E}\,\varepsilon_1=0$ and $\mathbb{E}\,\varepsilon_1^2=1$.…

Irreducibility conjecture. With probability $1-o(1)$, the characteristic polynomial of $M_n$ is irreducible.

Let $M$ be a real algebraic manifold of dimension $n$, let $s_d$ be a random section in the complex Fubini--Study model in dimension $n$ and codimension $r\in\{1,\dots,n\}$, and le…

Let $\mathcal N_n$ and $\mathcal L_n$ denote the relevant families of Newman and Littlewood polynomials of degree $n$, and let $N$ and $\widetilde N$ count zeros in the open unit d…

Persistence exponent conjecture. The persistence probability satisfies