31 problems

Simple-roots conjecture. For almost all polynomials $p$, the polynomial $\Delta_{\theta,h}(p)$ has only simple roots, apart from the stated exclusion.

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…

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…

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…

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

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…

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.

Let $L_n(X)$ and $L_{n,k}(X)$ be the polynomial families defined earlier in the paper, and let the roots of a polynomial be ordered decreasingly when indexed by $j$. For $k\leq 0$,…

Let $a_n(x)$ be the irreducible factor occurring in the conjectured factorization of $A_n(x)$. Brent's root-location conjecture. If $n>1$, $a_n(\rho)=0$, and $\rho\neq\pm2$, then ……

Let $p$ be a polynomial of degree $d$, rescaled so that all its roots lie in the unit disk. For $r>1$, let $\partial D_r(0)$ be the circle of radius $r$ centered at the origin, and…

Zero-strip-decreasing conjecture. The operator $\Delta_{\theta,h}$ defined on $\mathbb{C}[z]$ is zero strip decreasing.

Multiple-root conjecture. The polynomial $g_T(x)$ has multiple roots if and only if one of the following holds: (1) $T=(n-4)$ for $n\geq4$, with…

Let the roots of the corresponding univariate polynomials be all distinct, and let $\mathbf{a}$ be a vector whose entries are points in the complex plane. Let…

Disk conjecture. If

Let $f(x)$ be a polynomial of degree $n$. For a prime $p$ in $Spl(f)$, number the roots of $f(x)$ modulo $p$ as … For a subset $S\subseteq\{1,2,\dots,n\}$, write $Pr(f,S)$ for the…

Let $G=(V,E)$ be a connected graph, and let $\mathrm{Rel}(G;q)$ denote its all-terminal reliability polynomial in the edge-failure probability $q$. A complex number $z$ is an all-t…

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 $G=(V,E)$ be a simple graph, let … be its total domination polynomial, where $n$ is the order of $G$ and $d_t(G,i)$ counts the total dominating sets of cardinality $i$. A root…

Conjectural tropical analog of Descartes' rule of signs. For any real univariate polynomial $P(x)$, the number of its positive (negative) roots does not exceed the number of positi…