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Hilbert Series and Logarithmic Degrees of A-Hypergeometric Series
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.
Related
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- Hierarchical Melnikov Realization: Logarithmic Factor Improvement to Hilbert Number Lower Bounds
BibTeX
@misc{nagamine2026hilbert,
title = {{Hilbert Series and Logarithmic Degrees of $A$-Hypergeometric Series}},
author = {Mao Nagamine},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.01778},
eprint = {2608.01778},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.01778},
}
RIS
TY - UNPB
TI - Hilbert Series and Logarithmic Degrees of A-Hypergeometric Series
AU - Mao Nagamine
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2608.01778
UR - https://arxiv.org/abs/2608.01778
AB - 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.
ER -