14 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

Let $q$ be a fixed prime power. An infinite family of toric codes is a sequence $\{P_i\}_i$ of nonempty integral convex polytopes satisfying … with $n_i\to\infty$ as $i\to\infty$,…

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

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

Nonexistence conjecture. There are no good infinite families of toric codes.

Let $L$ be an $n$-dimensional integer lattice and consider the torus cellulated by unit hypercubes, with the relevant $q$-dimensional surfaces or chains representing nontrivial hom…

Let $L$ be an $n$-dimensional integer lattice, let $q=n/2$, and cellulate the associated torus using integer hypercubes, with qubits on the $q$-cells. Write $\gamma_{n,q}(L)$ for t…

Let ${\mathbb T}^n={\mathbb R}^n/\Lambda$ be an $n$-dimensional torus, where $\Lambda$ is a lattice. Consider the least-volume unoriented $n/2$-dimensional surface representing non…

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…

Consider the family $\mathcal{F}$ of polynomials … in $\mathbb{F}_p[u]$, where $p$ is prime. A polynomial has factorization pattern $\lambda_0=1^4$ when it splits completely into f…