6 problems

Let $G$ be a cycle graph with $\ell\geq 3$, let $R=E_{\mathbb{K}}[\operatorname{STAB}(G)]$ be its Ehrhart ring, and write its h-vector as $(h_0,h_1,\ldots,h_s)$ with $h_s\neq 0$. S…

Let $S_n:=\mathbb{Q}[y_1,\dots,y_n]$ with the degree-lexicographic term order $\prec$ satisfying $y_1\succ y_2\succ\dots\succ y_n$. Assume the polynomiality conjecture, so that eve…

Let $n\in\mathbb{N}$. There exist a $\mathbb{Z}^2$-graded polynomial ring $\Bbbk[x_1,\dots,x_m]$ for some $m\in\mathbb{N}$, a graded subalgebra $A_n\subseteq\Bbbk[x_1,\dots,x_m]$,…

Let $\mathbf{L}_n$ be the Lecture Hall cone, and let its Ehrhart ring be the graded algebra generated by lattice points of the cone. A toric deformation is described via initial su…

Let $\mathcal C$ be a uniform clutter with edge incidence vectors $v_1,\ldots,v_q$, and suppose its packing property holds. Define … as the matrix whose columns are the augmented v…

Let $\mathcal C$ be a uniform clutter with incidence vectors $v_1,\ldots,v_q$, and let $P$ be the associated polytope. The clutter satisfies the packing property if every minor sat…