65 problems

Ohsugi–Tsuchiya conjecture. The Ehrhart $h^*$-polynomials of symmetric edge polytopes are $\gamma$-positive.

Let $P$ be a lattice polytope in $\mathbb{R}^d$, and let $h^*_{rP}(z)$ denote its unweighted Ehrhart $h^*$-polynomial after dilation by the integer $r$. Beck–Stapledon conjecture.…

A classical symmetric edge polytope is the symmetric edge polytope associated with a graph, and its $h^\ast$-polynomial is the Ehrhart $h^\ast$-polynomial of that polytope. Gamma-p…

Golyshev's canonical line hypothesis. Every $d$-dimensional smooth Fano polytope with $d\leq 5$ satisfies the canonical line hypothesis. This connects the roots of Ehrhart polynomi…

Let $P$ be a $D$-dimensional Gorenstein Fano lattice polytope, and let $\alpha$ be a root of its Ehrhart polynomial. Ehrhart root strip conjecture. Every such root satisfies … This…

Let $B_n$ be the polytope of $n\times n$ doubly stochastic matrices, and let $e(B_n,t)$ denote the number of lattice points in $tB_n$. The dimension of $B_n$ is $(n-1)^2$. For a la…

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 $P\subset\mathbb{R}^d$ be a rational polytope, and write $i_P(t)=\#(tP\cap\mathbb{Z}^d)$ for its Ehrhart quasi-polynomial. For fixed $d$, the minimum Ehrhart period conjecture.…

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

Let $n$ be a positive integer, let $H_n(r)$ denote the number of $n\times n$ magic squares with line sums equal to $r$, and write … where $d=n^2-3n+2$ and $h_0=1$, $h_i=h_{d-i}$ fo…

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…

Fix $t\in\mathbb{N}$ and skew-shape data $\lambda/\mu$ arising from the generalized Pitman–Stanley polytope. Let $\operatorname{PP}_{k\lambda/k\mu}(t)$ denote the number of plane p…

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 $D$ be a cylindric shape and let $\beta$ be a weight. Write $\mathrm{CSSYT}(kD,k\beta)$ for the cylindric semistandard Young tableaux of shape $kD$ and weight $k\beta$. Cylindr…

Let $\operatorname{Pan}_{r,s,n}$ be a panhandle matroid, with $1\leq r\leq s<n$, and let $\mathcal{P}_{\operatorname{Pan}_{r,s,n}}$ be its matroid polytope. A lattice polytope is E…

Let $\mathsf{M}$ be a matroid, and let $\operatorname{ehr}(\mathsf{M},t)$ denote the Ehrhart polynomial of its matroid base polytope. Ehrhart coefficient conjectures. The coefficie…

Positivity conjecture for $i(\overline{\mathbf{C}}_n^{a,b},t)$. The polynomial