7 problems

Let $d$ and $n$ be positive integers, let $C^d_n$ be the cyclic $d$-polytope with $n$ vertices, and let $\mathsf{Inv}^d_{d+2}$ denote the algebra of degree-$d$ volume invariants fo…

Let $K$ be a triangulation of the $d$-simplex $\Delta^d$ associated with a $d$-dimensional cyclic polytope, meaning that it is obtained from the boundary of the polytope by removin…

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…

An integral cyclic polytope is a cyclic polytope with integral vertices. A lattice polytope is very ample when its associated semigroup is generated up to saturation in the usual t…

Let $C_d(\tau_1,\dotsc,\tau_n)$ and $C_d(\tau_1',\dotsc,\tau_n')$ be integral cyclic polytopes, and let $\Delta_{ij}$ denote the corresponding gaps used in the source. Assume that…

Let $C_4(\tau_1,\tau_2,\dotsc,\tau_n)$ be a cyclic polytope, and write $\Delta_{ij}=\tau_j-\tau_i$ for $i<j$. A lattice polytope is normal when every lattice point in every positiv…

A cyclic polytope is the convex hull of points on the moment curve; here, normal means that every lattice point in every positive dilate is a sum of lattice points of the polytope.…