111 problems

Let $\mu\in\mathcal P(\mathbb C)$ be a probability measure on $\mathbb C$, let $(X_i)_{i\in\mathbb N}$ be independent and identically distributed random variables with law $\mu$, a…

Let $A$ be a random reciprocal polynomial in the restricted coefficient model, with coefficients chosen independently and uniformly from … and condition on $A(0)=1$. Odlyzko–Poonen…

Shepp–Vanderbei's conjecture. Identified with its counting measure, this set converges as $n\to\infty$ to a Poisson point process. In particular, the closest root is typically at d…

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 $k\in\mathbb{N}$ be fixed, and consider the empirical measure of the zeros of the $k$-th derivative of the random polynomial described in the surrounding setup. Angst–Malicet–P…

Let a semi-regular sequence mean a sequence of homogeneous polynomials satisfying the semi-regularity condition. Pardue's conjecture. Semi-regular sequences are sequences of generi…

Let $f$ be a random complex polynomial of degree $d$, and let $c_1$ be the constant in the Renormalized Graeffe Iteration approximation theorem, under which $b\geq 1+\log_2 d$ and…

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…

Let $\{f_i\}_{i=1}^n$ be a functional system satisfying condition (d) with parameter $p\in[0,1/2)$. The source's random polynomials are those denoted by $F_n$ in equation, and Theo…

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

Let $f_n(x)=a_0+a_1x+\cdots+a_nx^n$ be a random polynomial with i.i.d. coefficients satisfying the hypotheses at the beginning of the section, and let $E_n$ denote its expected num…

Let $f_n(x)=a_0+a_1x+\cdots+a_nx^n$ be a random polynomial with nondegenerate, zero-mean, i.i.d. coefficients, and let $p_{n,k}$ denote the probability that $f_{n+1}$ has exactly…

Root-statistics universality conjecture. In the limit $N\to\infty$, the local statistics of the roots of polynomials in $\mathcal{C}_N$ are independent of the choice of joint proba…

Let $\mu_0$ be a rotationally invariant, compactly supported probability measure satisfying some mild assumptions, and let $P^N_0$ be any sequence of polynomials whose empirical ro…

Let $(m_n)_{n\geq 1}$ be a sequence of positive integers, and let $P_{n,m_n}$ be the structured polynomial whose roots lie on $n$ concentric circles with $m_n$ equidistributed poin…

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…

Small-order higher-derivative convergence conjecture. For every probability measure $\mu$ on the complex plane and every sequence $k_n$ satisfying $k_n=o(n)$, $\mu_{P_n^{(k_n)}}$ c…

Bogomolny–Bohigas–Leboeuf conjecture. The transition between crystalline and liquid-like patterns of distribution of zeros, as the second moments of the coefficients increase, is a…

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 $N_{I,\boldsymbol{\xi}}$ denote the number of real zeros in a natural interval $I$ for one of the random polynomial models considered in the paper. Suppose that the coefficient…

Let $f:\mathbb Z_{>0}\to\mathbb Z_{>0}$ be a function such that, for every $d$, there is at least one prime with $d<p\leq f(d)$. Choose a prime $p$ uniformly at random from…

Let $\mathbb{B}_{d,n}$ denote the large degree model of degree-$d$ hypersurfaces in projective $n$-space, and let $V\in\mathbb{B}_{d,n}$. Asymptotic smoothness conjecture. As…

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…

Bary-Soroker–Kozma's conjecture. With high probability as $n\to\infty$, the Galois group of $f_n$ is the full symmetric group; equivalently, $\Delta(f_n)$ is not a square.

Let $n$ and $m$ be the growing integer parameters defining the specific samplings of rotationally invariant measures used in the paper, and let the conclusion of Theorem denote the…