Mao Nagamine | Searcharxiv

arXiv2026

Hilbert Series and Logarithmic Degrees of $A$-Hypergeometric Series

Fix a generic weight vector and a fake exponent of a homogeneous $A$-hypergeometric system. Using all corresponding standard pairs, including embedded ones, we construct an Artinian quotient of the Stanley--Reisner ring of the link of the negative support. Its Hilbert series gives the graded dimensions of the orthogonal complement of the local fake indicial ideal and, under the Okuyama--Saito Frobenius condition, those of the leading logarithmic coefficient space of actual series solutions. The construction requires no Cohen--Macaulay hypothesis. When a top-dimensional standard pair occurs and the link is Cohen--Macaulay, the Hilbert series specializes to the $h$-polynomial of the link.

math.AC↗

arXiv2026

$A$-hypergeometric Series with Parameters in the Core

In this paper, we discuss the computational approach to the results established by Okuyama and Saito. Although their results are often difficult to compute, we prove that, when the negative support of a fake exponent $v$ with respect to a generic weight $w$ is included in a certain set, solutions can be computed using only the reduced Gröbner basis, and we can construct all $A$-hypergeometric series with exponent $v$ in the direction $w$ by Frobenius's method. As an example, we describe the Aomoto-Gel'fand system of type $3 \times 3$ in details.

math.AG↗