Core Coordinates and Negative Support in A-Hypergeometric Systems

Abstract

For a homogeneous $A$-hypergeometric system, let $K_v$ be the intersection of the negative support family for a fake exponent $v$, and let $C_w$ be the set of indices of variables that do not occur in any minimal monomial generator of ${\mathrm{in}}_w(I_A)$. We prove that $K_v=\mathrm{nsupp}(v)\cap C_w$ holds for every parameter and every fake exponent if and only if $\mathrm{Cone}(A_{C_w})$ meets the interior of $\mathrm{Cone}(A)$. We give equivalent criteria using the reduced Gröbner basis and the associated regular triangulation. When the condition fails, we construct an infinite family of fake exponents for which the equality fails.

arXiv comment16 pages