59 problems

Let $\mathcal{E}_d^{(s)}$ be the set of coefficient vectors of monic degree-$d$ real polynomials whose roots include exactly $s$ pairs of nonreal conjugates, and let … Here $d$ is…

Let $P$ be a lattice polytope and let $L_P(t)$ be its Ehrhart polynomial of degree $d$, with roots $\alpha_i$. Vertical strip conjecture. Every root satisfies … for all $i$. This c…

Let $M_d(t)$ be the polynomial associated with degree $d$, and let $\gamma_d$ be a root of $M_d(t)$ with largest norm. An SNN polynomial is a degree-$d$ polynomial satisfying Stanl…

Vavasis's conjecture. There exist two universal constants $0<\iota_1\leq 1\leq \iota_2$ such that the defined quantities satisfy the corresponding lower and upper bounds, uniformly…

Let $p$ be a degree-$n$ univariate complex polynomial, and let its DR-circles be the circles centered at the roots of $p'$ with the corresponding radii $\rho_j$. For a root $z_i$ o…

Let $p$ be a degree-$n$ univariate complex polynomial. Let its DR-circles be the circles centered at the roots of $p'$ with radii $\rho_j$, where … For each root of $p$, consider i…

Fast-convergence basin conjecture. There is a universal constant $\eta_1>0$ such that, for every simple root $z_i$, some DR-circle contains a segment of length $\eta_1/n$ radians l…

Annular root-location conjecture. The roots of $p$ lie in the union of these $n-1$ annuli. This would provide localized regions around derivative roots for finding all polynomial r…

Real-part bound. All roots $\alpha$ of Ehrhart polynomials of lattice $d$-polytopes satisfy

Majorization conjecture. If $T\in\mathcal{A}'$ and $n\ge 1$, then for every $P\in\mathcal{C}_n$ at least one of the relations

Root-location conjecture. All the roots of the polynomials $\alpha_l(m)$ and $\beta_l(m)$ lie on the line $\operatorname{Re}(m)=-1/2$. This conjecture is based on extensive numeric…

de Bruijn–Springer conjecture. The inequality

Let $K$ be a convex set in ${\bf R}^n$, and let $a_1\leq a_2\leq\cdots\leq a_n$ be the real parts of the roots of … Let $r$ and $R$ be the radii of $K$ relative to the $n$-dimensio…

Hypothesis R. Every null root of every polynomial over $(\mathcal A,\mathcal A_0)$ is a factor-root.

The coefficient-ratio conjecture. Except for countably many values of $a$, for every positive integer $l$, non-negative integer $m$, and integer $n$ satisfying $0<n\leq m+1$,

The root-concentration conjecture. Except for countably many values of $a$, for every positive integer $l$ and non-negative integer $m$, the maximal distance between the roots of…

Negative-index amplitude-bound conjecture. Numerical evidence indicates that

Negative-index root-pattern conjecture. Numerical calculations and symbolic manipulations indicate that

Let $G$ be a graph of order $n$, and let $z$ be a nonzero degree root of $G$ with argument $\arg(z) \in (-\pi,\pi]$. Modulus bound conjecture. The degree root satisfies … This conj…

Norm-order preservation conjecture. If $b_1(C)\geq b_1(L)$, then

Shapiro's shadow conjecture. The set $\Upsilon_P$ is a closed domain in the convex hull of the roots of $P$; all critical points of $P$ lie on its boundary; its boundary has no inf…

Kroó–Peherstorfer–Saff conjecture. Each function $p\mapsto x_{n,j}(p)$ is increasing on $(0,\infty]$ for $j=1,\dots,k$. The conjecture concerns how the roots of the unique $L^p$-ex…

Let $S_n(w)$ and $T_n(w)$ denote the principal polynomials associated with partition statistics, whose roots have arguments tending to equidistribution on the unit circle as…

Single-row root-at-minus-one conjecture. When $\lambda$ has a single row, the polynomials $a_\sigma^\lambda(n)$ have $n=-1$ as a root.

Ellipse conjecture. The roots of $f_n(\lambda)=F_{n+1}$ lie on an ellipse.