A sharp bound for hypergeometric rank in dimension three (Q6624723)

\(A\)-hypergeometric holonomic \(D\)-modules associated to an integer matrix \(A\) and a complex parameter \(\beta\), also known as GKZ-systems, were introduced in [\textit{I. M. Gel'fand} et al., Funct. Anal. Appl. 23, No. 2, 94--106 (1989; Zbl 0721.33006); translation from Funkts. Anal. Prilozh. 23, No. 2, 12--26 (1989); correction Funct. Anal. Appl. 27, No. 4, 295 (1989)] to generalize classical hypergeometric equations; see also the correction of the hypotheses of their result on the dimension of the solution spaces in [\textit{I. M. Gel'fand} et al., Funct. Anal. Appl. 27, No. 4, 1 (1993; Zbl 0994.33501); translation from Funkts. Anal. Prilozh. 27, No. 4, 91 (1993)]. This dimension is known as the holonomic rank of the system and denoted by \(\mathrm{rank}(M_A(\beta))\).\N\NIn the Cohen-Macaulay case, \(\mathrm{rank}(M_A(\beta))\) equals the normalized volume \(\mathrm{vol}(A)\) with respect to the integer lattice. In the non-Cohen Macaulay case, it might happen that for special parameters \(\beta\), \(\mathrm{rank}(M_A(\beta))\) is greater than \(\mathrm{vol}(A)\). When the affine dimension of \(A\) is equal to \(1\), it was shown in [\textit{E. Cattani} et al., Duke Math. J. 99, No. 2, 179--207 (1999; Zbl 0952.33009)] that \(\mathrm{rank}(M_A(\beta))\) is always bounded by \(\mathrm{vol}(A) +1\). For any dimension \(d\) it was proven in [\textit{C. Berkesch} and \textit{M.-C. Fernández-Fernández}, Bull. Lond. Math. Soc. 54, No. 1, 182--192 (2022; Zbl 1520.13041)] that \(\frac{\mathrm{rank}(M_A(\beta)} {\mathrm{vol}(A)} \le (d-1) \) for \emph{ generic rank jump} parameters. However this quotient can be exponentially large for special values of \(\beta\). In the paper under review the authors show that if \(A\) is a \textit{pointed} configuration and \(d=3\), the quotient is always strictly smaller than \(d-1=2\). The proof of this result uses tools from Ehrhart theory.