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The dimension of the space of holomorphic solutions at nonsingular points (also called the holonomic rank) of a $A$--hypergeometric system $M_A (\beta)$ is known to be bounded above by $ 2^{2d}\operatorname{vol}(A)$, where $d$ is the rank of the matrix $A$ and $\vol (A)$ is its normalized volume. This bound was thought to be very vast because it is exponential on $d$. Indeed, all the examples we have found in the literature verify that $\operatorname{rank}(M_A (\beta)) 1$.

Exponential growth of rank jumps for A-hypergeometric systems