$A$-hypergeometric Series with Parameters in the Core
Mao Nagamine
TL;DR
The paper develops a practical framework for constructing logarithmic $A$-hypergeometric series in nongeneric parameter regimes by extending Okuyama–Saito via Frobenius’s method. Under a generic weight $\bm{w}$ and appropriate control of the fake exponents’ negative support, the solution space can be generated from the reduced Gröbner basis, with a basis obtained from $P^{\perp}$ acting on a Frobenius-perturbed series $F(\bm{x},\bm{s})$. In the unimodular triangulation setting and when $\bm{\beta}$ lies in the core, the space of $A$-hypergeometric series in direction $\bm{w}$ is explicitly constructible, and the method yields concrete bases in examples. The authors illustrate the approach on the Aomoto–Gel'fand system of type $3\times3$ and apply the framework to K3-surface period problems, highlighting computational advantages over limit-based constructions. Together, these results provide both algorithmic and conceptual tools for analyzing nongeneric $A$-hypergeometric systems in geometric contexts.
Abstract
In this paper, we discuss the computational approach to the results established by Okuyama and Saito \cite{log19}. Although their results are often difficult to compute, we prove that, when the negative support of a fake exponent $\bm{v}$ with respect to a generic weight $\bm{w}$ 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 $\bm{v}$ in the direction $\bm{w}$ by Frobenius's method. As an example, we describe the Aomoto-Gel'fand system of type $3 \times 3$ in details.