Exponential growth of rank jumps for <var>A</var>-hypergeometric systems

  • María-Cruz Fernández-Fernández

    Universidad de Sevilla, Spain


The dimension of the space of holomorphic solutions at nonsingular points (also called the holonomic rank) of an AA-hypergeometric system MA(β)M_A (\beta ) is known to be bounded above by 22dvol(A)2^{2d}\operatorname{vol}(A), where dd is the rank of the matrix AA and vol(A)\operatorname{vol}(A) is its normalized volume. This bound was thought to be much too large because it is exponential in dd. Indeed, all the examples we have found in the literature satisfy rank(MA(β))<2vol(A)\operatorname{rank}(M_A (\beta ))<2 \operatorname{vol}(A). We construct here, in a very elementary way, some families of matrices A(d)Zd×nA_{(d)}\in \mathbb{Z}^{d \times n} and parameter vectors β(d)Cd\beta_{(d)} \in \mathbb{C}^d, d2d\geq 2, such that rank(MA(d)(β(d)))advol(A(d))\operatorname{rank} ( M_{A_{(d)}} (\beta_{(d)} ))\geq a^d \operatorname{vol}(A_{(d)}) for some a>1a>1.

Cite this article

María-Cruz Fernández-Fernández, Exponential growth of rank jumps for <var>A</var>-hypergeometric systems. Rev. Mat. Iberoam. 29 (2013), no. 4, pp. 1397–1404

DOI 10.4171/RMI/761