7 problems

Authors–Shibata conjecture. The following conditions are equivalent:

Let $P$ be a poset, let $\widehat{P}$ be its augmented poset, let $\mathcal{Q}(\widehat{P})$ denote the set of maximal chains of $\widehat{P}$, and let $\mathfrak{C}_P$ and…

Let $C_{2s+1}$ be the cycle graph on $2s+1$ vertices, and let $K[C_{2s+1}]$ denote its toric ring. Write its $h$-vector as a sequence with entries $h_1,h_2,\ldots,h_{s-1}$ appearin…

Let $\mathcal{P}$ be a $(0,1)$-polytope, and let $k[\mathcal{P}]$ be the toric ring generated by $\mathcal{P}$. A ring $R$ is trivial if it can be constructed from polynomial rings…

Normality–$K_5$-minor conjecture. The following conditions are equivalent:

Let $P$ be a finite poset that is a disjoint union of chains. Write ${\mathcal J}(P)$ for the poset of order ideals of $P$, and let $K[\mathcal O(P)]$, $K[\mathcal C(P^)]$, and…

Let $Q$ be the affine semigroup associated with the cyclic polytope construction in the paper, and let $K[Q]$ be its semigroup algebra over the field $K$. Write $n$ and $d$ for the…