Regularity of Symbolic Powers of Polymatroidal Ideals and Supermodular Prime Intersections
Abstract
Let S=𝕂[x1,…,xn] be a polynomial ring over a field 𝕂. For a polymatroidal ideal I⊆ S generated in degree d, we prove ω(I(s))=reg I(s)=reg Is=sd (s≥1), where symbolic powers use all associated primes and ω denotes the largest degree of a minimal generator. Thus every symbolic power has a degree resolution. The argument uses a regularity bound for supermodular prime intersections, obtained from Takayama's formula and a homological estimate for noncover complexes. For every nonzero proper squarefree monomial ideal L, we also show reg L(s)≤ smin{m,λ(L)−htL+1}, where m counts the minimal primes and λ(L) is the support size, and obtain equality for skeleton and pure skeleton ideals. Finally, for every d≥3, we construct a simple graphic matroid whose basis ideal is generated in degree d and has a second symbolic power that is not componentwise linear.
Related
- Regularity of symbolic powers of complementary edge ideals
- Comparisons between ordinary and symbolic powers of edge ideals with respect to regularity and depth
- Regularity, depth and Serre depth of symbolic powers of complementary edge ideals
- Uniform bounds on symbolic powers in regular rings
- ARITHMETIC UNLIKELY INTERSECTIONS IN POWERS OF THE MULTIPLICATIVE GROUP
BibTeX
@misc{ahmed2026regularity,
title = {{Regularity of Symbolic Powers of Polymatroidal Ideals and Supermodular Prime Intersections}},
author = {Chwas Ahmed and Mohammed Rafiq Namiq},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.38655},
eprint = {2610.38655},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.38655},
}
RIS
TY - UNPB
TI - Regularity of Symbolic Powers of Polymatroidal Ideals and Supermodular Prime Intersections
AU - Chwas Ahmed
AU - Mohammed Rafiq Namiq
PY - 2026
DA - 2026/10/28
PB - arXiv
DO - 10.48550/arxiv.2610.38655
UR - https://arxiv.org/abs/2610.38655
AB - Let S=𝕂[x_1,…,x_n] be a polynomial ring over a field 𝕂. For a polymatroidal ideal I⊆ S generated in degree d, we prove ω(I^(s))=reg I^(s)=reg I^s=sd (s≥1), where symbolic powers use all associated primes and ω denotes the largest degree of a minimal generator. Thus every symbolic power has a degree resolution. The argument uses a regularity bound for supermodular prime intersections, obtained from Takayama's formula and a homological estimate for noncover complexes. For every nonzero proper squarefree monomial ideal L, we also show reg L^(s)≤ smin{m,λ(L)−htL+1}, where m counts the minimal primes and λ(L) is the support size, and obtain equality for skeleton and pure skeleton ideals. Finally, for every d≥3, we construct a simple graphic matroid whose basis ideal is generated in degree d and has a second symbolic power that is not componentwise linear.
ER -