7 problems

Let $\mathcal{C}(E,D,X)$ be the toric code associated to the $1$-cycle $E$, the $T$-invariant Cartier divisor $D$, and the toric variety $X$. Assume that $X$ is a nonsingular toric…

Toric-code degree conjecture. If $q$ is sufficiently large, then every $f\in L(G)=H^0(X,\operatorname{\mathcal O}(G))$ has at most $n=\operatorname{deg}(P)$ zeros in…

Hansen's conjecture. If $q$ is sufficiently large, then every $f\in L(G)=H^0(X,\operatorname{\mathcal O}(G))$ has at most $n$ zeros in $X(\mathbb{F}_q)$, and consequently

Dolorfino et al.'s bounded-hypercube conjecture. If $\{M(P_i)\}_i$ is bounded, then

Bounded-Minkowski-length conjecture. If the sequence $\{L(P_i)\}$ is bounded, then

Bounded-hypercube conjecture. If the sequence $\{M(P_i)\}$ is bounded, then

Let $P_6^{(4)}$ and $P_6^{(5)}$ be the polytopes indexing the corresponding six-dimensional toric codes $C_{P_6^{(4)}}$ and $C_{P_6^{(5)}}$ over $\mathbb{F}_8$. The conjecture. The…