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When Hilbert functions determine four-variable generic initial ideals
Abstract
Over a field of characteristic zero, we characterize the Hilbert functions that determine the degree reverse lexicographic generic initial ideal among Artinian quotients in four variables with the $2$-strong Lefschetz property. Our criterion is a finite numerical test that strictly extends the quasi-symmetry condition of Harima and Wachi. The proof parametrizes the possible generic initial ideals by flags of lower sets in the Borel orders on their common three-variable section. We also give nonrecursive formulas for the standard monomials and minimal generators of the almost reverse lexicographic ideal, and specialize them to generic complete intersections in at most four variables without restrictions on the degrees.
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BibTeX
@misc{trung2026when,
title = {{When Hilbert functions determine four-variable generic initial ideals}},
author = {Van Duc Trung},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.07799},
eprint = {2610.07799},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.07799},
}
RIS
TY - UNPB
TI - When Hilbert functions determine four-variable generic initial ideals
AU - Van Duc Trung
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2610.07799
UR - https://arxiv.org/abs/2610.07799
AB - Over a field of characteristic zero, we characterize the Hilbert functions that determine the degree reverse lexicographic generic initial ideal among Artinian quotients in four variables with the $2$-strong Lefschetz property. Our criterion is a finite numerical test that strictly extends the quasi-symmetry condition of Harima and Wachi. The proof parametrizes the possible generic initial ideals by flags of lower sets in the Borel orders on their common three-variable section. We also give nonrecursive formulas for the standard monomials and minimal generators of the almost reverse lexicographic ideal, and specialize them to generic complete intersections in at most four variables without restrictions on the degrees.
ER -