39 problems

Let $P$ be a $d$-polytope with vertex set $V$, and let $d'\le d-1$. Suppose that for every integer $k$ with $0\le k\le d'$, condition (dfcon) is satisfied. Write $i(P,m)$ for the E…

Let $\mathcal{GT}(\lambda,\sigma)$ be the generalized Gelfand–Tsetlin polytope associated with a partition $\lambda$ and a permutation $\sigma$, and let…

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…

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

For $\mathbf{b}=(b_1,\dots,b_n)\in\mathbb{Z}_{>0}^n$, let $\mathfrak{X}_n(\mathbf{b})$ denote the $\mathbf{b}$-parking-function polytope, the convex hull of all $\mathbf{b}$-parkin…

Let $a,b$ be positive integers. Let ${\texttt{Pan}}_{b+1,a+b,a+b+2}$ be the panhandle matroid, let $i(M,t)$ denote the Ehrhart polynomial of the matroid polytope of $M$, and let…

The Ehrhart–Zeta duality conjecture. For every linear arbor $t$,

The real-rootedness conjecture for arbor Ehrhart polynomials. For any arbor $t$, all roots of $E_t(u)$ are negative real numbers in the interval $[-1,0]$, excluding $0$.

Let $n$ be any positive integer and let $A_0$ be an alcove in the hypersimplex $\Delta_{2,n}$. Let $\mathcal{P}_{A_0}=(V,E)$ be the breadth-first search order of $\Gamma_{2,n}$ root…

Let $\mathbf{w}$ be a generalized snake word, let $\varphi_i(\mathbf{w})$ be the word obtained by the swap operator, and write $L(\mathbf{w};t)$ for the Ehrhart polynomial associat…

Let $\mathbf{w}$ be a generalized snake word of length $n+1$. Let $h^*(\mathbf{w};z)$ denote its $h^*$-polynomial, and let $L(\mathcal{O}(P(\mathbf{w}));t)$ denote the Ehrhart poly…

Cross-degree and interlacing conjecture. The inequalities

Higashitani–Kummer–Michałek conjecture. (i) For any complete multipartite graph $K_{a_1,\dots,a_k}$, the roots of $E_{a_1,\dots,a_k}$ lie on $\operatorname{CL}$. (ii) If…

Let $\mathsf{M}$ be a matroid, and let $\mathscr{P}(\mathsf{M})$ be its matroid base polytope. Denote by $\operatorname{ehr}(\mathscr{P}(\mathsf{M}),t)$ its Ehrhart polynomial. Ehr…

For any $n\geq 1$, let $\operatorname{Tes}_{n}(\boldsymbol{1})$ and $\operatorname{Tes}_{n}(1,0,\dots,0)$ be Tesler polytopes, where $\boldsymbol{1}=(1,\dots,1)$. A polytope is Ehr…

Let $P$ be a lattice polytope that is integrally closed (IDP), meaning that for every $m\in\mathbb{Z}_{>0}$ and every $x\in mP\cap\mathbb{Z}^n$ there exist…

Let $a_0=1$, $a_1=2$, and define $a_n=3a_{n-1}-a_{n-2}$. For ${\boldsymbol{i}}=(i_1,i_2)\in\langle a_n\rangle\times\langle a_{n+1}\rangle$, let $u(\alpha({\boldsymbol{i}}))$ be the…

Truncated Ehrhart-like polynomial conjecture. For every $\lambda\in P$ and every good $\mathbf{k}\in\mathbb{N}[\Phi]^W$,

Let $\textup{Car}_{n+1}$ be the Caracol graph on $n+1$ vertices, and let $E_{\textup{Car}_{n+1}}(x)$ denote the Ehrhart polynomial associated with its flow polytope. Caracol Ehrhar…

For $\boldsymbol{a}\in\mathbb{N}^n$, let $\operatorname{Tes}_n(\boldsymbol{a})$ be the polytope of upper triangular $n\times n$ matrices with nonnegative entries and hook sum…

The Birkhoff polytope $B_n$ is the convex hull of all $n\times n$ permutation matrices, equivalently the set of real nonnegative $n\times n$ matrices whose row and column sums are…

For a positive integer $r$ and a positive integer dimension $d$, define $\boldsymbol{q}_r=(r-1,(r-1)r,(r-1)r^2,\dots,(r-1)r^{d-1})$ and let the base-$r$ $d$-simplex be … Here…

Let $G$ be a complete $k$-partite graph of type $(a_1,\dots,a_k)$, and let $H_{a_1,\dots,a_k}$ denote its Ehrhart polynomial. Complete multipartite Ehrhart conjecture. The polynomi…

Let $(P_j)_{j=0}^{\infty}$ be a family of reflexive polytopes whose Ehrhart polynomials $f_j(x)=H_{P_j}(x)$ form an orthogonal polynomial sequence and satisfy … with…