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

Mao Nagamine

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arXiv · 2608.01778

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

Mao Nagamine

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Abstract

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.

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Mao Nagamine. 2026-08-03. Hilbert Series and Logarithmic Degrees of $A$-Hypergeometric Series. https://arxiv.org/abs/2608.01778

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