$A$-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 Gr\"{o}bner 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.