24 problems

Let $n\geq 3$, let $h\in\mathbb{R}^n$ be the normalized Hermite root vector, and define … Let $W=\mathbf{1}^\perp=\{u\in\mathbb{R}^n:u^\top\mathbf{1}=0\}$ be the mean-zero subspace…

Characteristic-polynomial conjecture. All roots of $f(x)$ are real numbers. If $m=n+2$, the points are in general position in $H^n$, and exactly $s$ of the $\alpha_i$ are positive,…

Let $p$ be a real-rooted polynomial of degree $k$, and let $M(p)$ denote its mesh, with $M(p)\geq 1$. Discrete Laguerre inequality. Then … This conjecture proposes a discrete analo…

For integers $q>0$ and $m=2,3$, let $H_n^{[m]}(q,t)$ denote the $(q,t)$-Hoggatt sums. The Sturm-sequence conjecture. The polynomial sequence … forms a Sturm sequence. The authors v…

Recursive decomposition inequality. For every relevant index $i$, one has $h_i\leq g_i$.

Let $W_{n,k}(x)=\sum_{m=0}^{k}w_{n,k,m}x^m$, where $w_{n,k,m}$ counts Dyck paths of semilength $n$ with $k$ occurrences of $UD$ and $m$ occurrences of $UUD$. For real-rooted polyno…

Let $W_{n,k}(x)=\sum_{m=0}^{k}w_{n,k,m}x^m$, where $w_{n,k,m}$ counts Dyck paths of semilength $n$ with $k$ occurrences of $UD$ and $m$ occurrences of $UUD$. A sequence of real-roo…

Real-rootedness and Eulerian interlacing conjecture. The polynomial $h_{\mathcal L}(x)$ has only real roots and is interlaced by the Eulerian polynomial $A_n(x)$ for every geometri…

Positivity and real-rootedness conjecture. For all $k,n\in\mathbb{N}$, $M_{\textnormal{id}_k,n}$ is a nonnegative linear combination of irreducible symmetric group characters. Furt…

For a positive integer $n$, define the Narayana polynomial … Given a composition $\mathbf q=(q_0,\ldots,q_k)\in\mathbb Z_{>0}^{k+1}$, define … An integral binary nestohedron…

Monotonicity conjecture. If $T$ is not monotone, then $T$ is not hyperbolicity preserving.

Jensen-polynomial zero-bound conjecture. For every such $p$ and every relevant index $i$,

Even-degree weighted Hawaiian conjecture.

Weighted Hawaiian conjecture.

Forsgård–Shapiro coefficient-difference conjecture. The number of real zeros of $f$ does not exceed $v(f)$.

Forsgård–Shapiro tropical conjecture. The number of real zeros of $f$ does not exceed the number of corners of the continuous piecewise-linear function $f_{\operatorname{trop}}$ on…

Brändén–Krasikov–Shapiro conjecture. The operator $T$ preserves the set of real-rooted polynomials of degree at most $m$ whose mesh is at least $1$ if and only if $T((x)_m)$ is rea…

Let $T$ be a constant-coefficient difference operator … It acts on hyperbolic polynomials of degree at most $m$ whose mesh is at least one, where the mesh is the minimum distance b…

Let $\{\alpha_i\}_{i=0}^\infty$ be a non-trivial sequence, meaning it is not zero for all but at most two indices, and suppose that $\alpha_i>0$ for some $i$. A discrete multiplier…

Let $\mathcal{L}\text{-}\mathcal{P}_n$ denote the real polynomials of degree $n$ with only real zeros, let $\mu(p)$ be the mesh size, and let $m$ denote Lebesgue measure. Discrete…

Let $\mathcal{L}\text{-}\mathcal{P}_n$ denote the real polynomials of degree $n$ with only real zeros, and let $\mu(p)$ be the minimum spacing between neighboring zeros. For $p$, d…

Let $\mathcal{L}\text{-}\mathcal{P}_n$ denote the real polynomials of degree $n$ with only real zeros, let $\mu(p)$ be their mesh size, and define … Rescaled discrete Laguerre conj…

Let $\mathcal{L}\text{-}\mathcal{P}_n$ denote the real polynomials of degree $n$ with only real zeros. For such a polynomial $p$, let $\mu(p)$ be the minimum spacing between neighb…

Su and Wang's real-rootedness conjecture. The polynomial