arXiv subjects

Go Okuyama

Publications and source records attributed to Go Okuyama.

5 recordsLinked to original sources

arXiv2026

Finite Directional Coefficient Modules over Euler Fibers of $A$-Hypergeometric Systems

Let $A$ be a full-rank integer matrix, let $\beta$ be a complex parameter, and consider the associated $A$-hypergeometric system. Without assuming homogeneity, pointedness, or positivity, we study directed formal logarithmic solutions over all exponent lattices in the Euler fiber. For a fixed lattice-generic direction, negative-support strata are described by rational sign polyhedra, and normalization reduces the coefficient equations on each exponent lattice to a finite directional coefficient module. We prove that only finitely many exponent lattices contribute nonzero directed solution spaces and that all contributing coefficient modules have finite length. Macaulay inverse-system duality identifies the dimension of the all-lattice solution space with the length of the corresponding all-lattice module. We also give an explicit formal-series realization and a polynomial colon-ideal criterion for all realizable lowest exponents.

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arXiv2025

Fourier-Orbit Construction of GKZ-Type Systems for Commutative Linear Algebraic Groups

We study GKZ-type D-modules arising from the actions of commutative linear algebraic groups G = TU (where T is a torus and U is unipotent) on a vector space. Building on Hotta's equivariant D-module framework, we formalize a Fourier-orbit construction that recovers the classical toric GKZ system and extends it to mixed torus-unipotent settings. We prove generic holonomicity via a parameter-free symbolic moment ideal and introduce two symbolic tools - the tp-envelope and the symbolic cap - for effective rank analysis and, under mild regularity, exact rank computation. A torus slice yields an explicit lower bound by the normalized lattice volume, explaining sharpness in the pure torus case. Examples exhibit irregular (Airy-type) behavior and resonant non-holonomicity, highlighting new phenomena beyond the toric setting.

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arXiv2025

Logarithmic $A$-hypergeometric series ${\textrm I}\! {\textrm I}\! {\textrm I}$

We study the logarithmic coefficients that can occur at a fixed fake exponent in an $A$-hypergeometric series subject to prescribed negative support conditions. Let $L=\operatorname{Ker}_{\mathbb Z}(A)$. For a fixed generic weight $w$, a fixed fake exponent $v_0$, and an ordered negative support family, we first derive a finite system of constant-coefficient differential equations whose solutions encode the admissible logarithmic coefficients. A normalization of the coefficient equations shows that, for $u\in L$, the normalized coefficient associated with $x^{v_0+u}$ depends only on the negative support of $v_0+u$. From this finite system, we identify the annihilator of the coefficient space with an explicitly defined colon ideal. For the negative support family determined by the direction $w$, we further identify this colon ideal with the primary component of the indicial ideal supported at $v_0$, shifted to the origin. We next introduce an ambient perturbation construction in which the fake exponent is perturbed in the full ambient space rather than only within the affine space $v_0+L_{\mathbb C}$. We prove that the ambient perturbation construction produces $A$-hypergeometric series and realizes the full coefficient space. Finally, we compare the ambient perturbation construction with the intrinsic perturbation construction developed in our previous papers. The intrinsic construction always yields a subspace of the full coefficient space, and it realizes the full coefficient space if and only if a natural equality between the corresponding colon ideals holds. We also give several sufficient conditions for this equality.

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arXiv2022

Logarithmic A-hypergeometric series II

In this paper, following [6], we continue to develop the perturbing method of constructing logarithmic series solutions to a regular A-hypergeometric system. Fixing a fake exponent of an A-hypergeometric system, we consider some spaces of linear partial differential operators with constant coefficients. Comparing these spaces, we construct a fundamental system of series solutions with the given exponent by the perturbing method. In addition, we give a sufficient condition for a given fake exponent to be an exponent. As important examples of the main results, we give fundamental systems of series solutions to Aomoto-Gel'fand systems and to Lauricella's FC systems with special parameter vectors, respectively.

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arXiv2011

The Freeness and Minimal Free Resolutions of Modules of Differential Operators of a Generic Hyperplane Arrangement

Let A be a generic hyperplane arrangement composed of r hyperplanes in an n-dimensional vector space, and S the polynomial ring in n variables. We consider the S-submodule D(m)(A) of the nth Weyl algebra of homogeneous differential operators of order m preserving the defining ideal of A. We prove that if n \geq 3, r > n,m > r - n + 1, then D(m)(A) is free (Holm's conjecture). Combining this with some results by Holm, we see that D(m)(A) is free unless n \geq 3, r > n,m < r - n + 1. In the remaining case, we construct a minimal free resolution of D(m)(A) by generalizing Yuzvinsky's construction for m = 1. In addition, we construct a minimal free resolution of the transpose of the m-jet module, which generalizes a result by Rose and Terao for m = 1.

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