3 problems

Let $A$ be a homogeneous matrix, let $I_A$ be its toric ideal, and let $H_A(\beta)$ be the $A$-hypergeometric system with parameter vector $\beta$. Its holonomic rank is the dimens…

Sturmfels's conjecture. The exceptional set is empty if and only if $I_A$ is Cohen–Macaulay. For Cohen–Macaulay $I_A$, the rank equals $\operatorname{vol}(A)$ for every parameter,…

Gevrey-slope conjecture. The map $\beta\mapsto {\operatorname{d}_s}(A,\beta)$ is upper-semicontinuous. Moreover,