Obstructions to intrinsic perturbation for $A$-hypergeometric series

Abstract

Okuyama--Saito ask whether, for an ordered negative support family at a fake exponent of an $A$-hypergeometric system, intrinsic perturbation inside the relation lattice can produce a strictly smaller coefficient space than ambient perturbation. It can, and we measure the gap by an intrinsic-perturbation obstruction module, a quotient of two colon ideals, whose degreewise dual is the ambient coefficient space modulo the intrinsic one. We realize this module by two right-exact sequences involving $\Tor_1$ and present it finitely by two antichains. For a homogeneous system, we present as a finite-dimensional cokernel the quotient of the canonical formal solution space by the span of the intrinsic series, taken over all fake exponents occurring in the canonical formal solution space and all ordered negative support families, and we compute the codimension of that span; the presentation and the codimension transfer to the holomorphic solutions on a common nonsingular domain. Configurations of lattice rank 1 and 2 with nonzero obstruction include one whose distinguished collection is ordered; that example settles the case Okuyama--Saito leave open. For every integer $q\geq1$ there is a configuration of lattice rank 2, with a connected column matroid and with a normal affine semigroup generated by the reduced Gröbner basis vectors, whose obstruction module has dimension $q^2$ and whose intrinsic holomorphic solutions have codimension at least $\lceil3q^2/4\rceil$; this codimension is unbounded at fixed lattice rank.

Disclosure

“This work was supported by JST SPRING, Grant Number JPMJSP2119. Large language models were used in the preparation of this paper. Codex was used to search for the configuration of Theorem 4.17 and to recompute its data independently, and Claude was used to draft and to revise the exposition. Every mathematical statement was checked by the author against the definitions and the cited results, and the author is responsible for the”

PDF page 41

Classification

Suggesting mathematical examples or conjectures

Multiplier

6

Verified

2026-08-23T05:22:20.368309Z

Structural counts