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Maximal shifts below the Taylor bound
Abstract
Let I⊆ S be a monomial ideal with minimal generator degrees between e and d, where 2≤ e≤ d. We prove that if ta(S/I)<ea, then ta+b(S/I)≤ ta(S/I)+ ⌈((2d−1)b)/2⌉ for a≥2, b≥0, and a+b≤pdS(S/I). Consequently, ta+b(S/I)≤ ae+db−1−⌊b/2⌋. In particular, if t2(S/I)<2e, then ti(S/I)≤ ⌈((2d−1)i)/2⌉−2(d−e) for 2≤ i≤pdS(S/I). We also obtain a regularity bound and give a quadratic family in which our estimate for the last shift is smaller than every bound obtained from ordinary subadditivity. The constant 2d−1 is sharp for d=2.
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BibTeX
@misc{abedelfatah2026maximal,
title = {{Maximal shifts below the Taylor bound}},
author = {Abed Abedelfatah},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.32135},
eprint = {2609.32135},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.32135},
}
RIS
TY - UNPB
TI - Maximal shifts below the Taylor bound
AU - Abed Abedelfatah
PY - 2026
DA - 2026/09/26
PB - arXiv
DO - 10.48550/arxiv.2609.32135
UR - https://arxiv.org/abs/2609.32135
AB - Let I⊆ S be a monomial ideal with minimal generator degrees between e and d, where 2≤ e≤ d. We prove that if t_a(S/I)<ea, then t_a+b(S/I)≤ t_a(S/I)+ ⌈((2d−1)b)/2⌉ for a≥2, b≥0, and a+b≤pd_S(S/I). Consequently, t_a+b(S/I)≤ ae+db−1−⌊b/2⌋. In particular, if t_2(S/I)<2e, then t_i(S/I)≤ ⌈((2d−1)i)/2⌉−2(d−e) for 2≤ i≤pd_S(S/I). We also obtain a regularity bound and give a quadratic family in which our estimate for the last shift is smaller than every bound obtained from ordinary subadditivity. The constant 2d−1 is sharp for d=2.
ER -