33 problems

Let $P$ be a poset, and let $W_P(t)$ denote its $W$-polynomial, also called its $P$-Eulerian polynomial. Neggers–Stanley conjecture. The polynomial $W_P(t)$ is real rooted. This co…

Ferroni–Matherne–Vecchi conjecture. The Chow polynomial $H_P(t)$ is real-rooted.

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…

Even-degree parameter-set conjecture. For all $k\in\mathbb{N}$, there exist constants $c_{2k}$ such that

Lee–Vindas-Meléndez–Wang conjecture. The Ehrhart $h^*$-polynomial $h^*(\mathcal{O}(P),t)$ is real-rooted. This conjecture concerns the root structure of Ehrhart $h^*$-polynomials f…

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…

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 $a_{-1}=a_{n+1}=0$, and let $f(x)=\sum_{k=0}^n a_kx^k$ be a polynomial whose zeros are all real and negative. Fisk–McNamara–Sagan–Stanley conjecture. The polynomial … also has…

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…

Let $u(t,x)$ denote the density evolution associated with repeatedly differentiating a real-rooted polynomial, as in the polynomial-root/free-convolution correspondence. The semici…

Let $f$ be a real-valued polynomial of degree $n$, let $m$ be an integer satisfying $0\leq 2m\leq n$, and write $f^{(j)}$ for the $j$th derivative of $f$. Foster–Krasikov conjectur…

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

Let $\Phi$ be an irreducible root system of type $A_n$, and let $\Gamma(\Phi)$ be its cluster subdivision. Its local $h$-polynomial is … where $I$ indexes the simple roots,…

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

Even-degree weighted Hawaiian conjecture.

Weighted Hawaiian conjecture.

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…

Forsgård–Kostov–Shapiro conjecture. For every sign pattern, every admissible pair with positive $pos$ and $neg$ is realizable by a polynomial having that sign pattern; equivalently…