8 problems

Positive-root distribution conjecture. There are exactly $N-k$ positive roots in $x$ when

Let $P_n$ be the path graph on $n$ vertices, let $f_n(\lambda)$ be its characteristic polynomial, and let $R_n$ be the set of roots of $f_n(\lambda)=F_{n+1}$, where $F_n$ is the Fi…

Let $P_n$ be the path graph on $n$ vertices, let … be its characteristic polynomial, and let $F_0=0$, $F_1=1$, with $F_n$ denoting the Fibonacci numbers. For each $n$, let $R_n$ be…

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…

Bounded-coefficient limit-ratio conjecture. If the limit ratio of such a sequence is strictly below some constant $c(N)>0$, then its limit ratio is equal to $0$.

Let $P$ be a lattice $D$-polytope, and let its Ehrhart polynomial have complex roots. For a root $\alpha$, write $\Re(\alpha)$ for its real part. Beck–De Loera–Develin–Pfeifle–Stan…

Let $P$ be a Gorenstein Fano polytope of dimension $D$, and let $\alpha$ be a root of its Ehrhart polynomial. Gorenstein Fano Ehrhart root narrow-strip conjecture. All roots…

Let $P$ be a lattice polytope of dimension $D$, and let its Ehrhart polynomial be the polynomial whose value at a nonnegative integer $m$ counts the lattice points in $mP$. Ehrhart…