Skip to content
DraconiticResearch
Search Sources API
194,161 studies31 sourcesUpdated 50 min ago
PreprintCC BY 4.0arXiv

Polynomial bounds for the stability of associated primes of monomial ideals

  • Tran Nam Trung
Open at sourcePDFDOICite

Abstract

Let I be a nonzero proper monomial ideal in k[x1,…,xr], and let d be the maximal degree of a minimal monomial generator. We prove that the index of stability of the associated primes of its ordinary powers satisfies astab(I)≤((4e)r−1)/4 dr−1 (r≥3), while astab(I)=1 for r=2. If I has s≥2 minimal generators, we also obtain astab(I)≤min{r,s}(s−1)dmin{r−1,s−2}. The first estimate answers Hoa's question on polynomial bounds. The proofs reduce stabilization to integer optimization for a matrix whose last row records the power of the ideal. We bound the degrees of its circuits and Graver basis elements and prove proximity estimates for optimal solutions in fixed fibers. A Steinitz rearrangement and a lattice-point count in the difference body of a simplex give the generator-independent bound. The matrix estimates hold for arbitrary indexed integral configurations in a simplex and real linear objective functions. The stability bounds hold over an arbitrary field, without normality or equigeneration assumptions.

Related

  1. On the existence of the maximal ideal in the set of associated primes of monomial ideals
    M. Cimpoeaş, M. Nasernejad, A. A. Qureshi
    arXivPreprintCC BY 4.0Full textAlgebra & number theoryarXiv
  2. On the Fluctuations of Associated Primes and Depths of Powers of Monomial Ideals
    Mircea Cimpoeas, Mehrdad Nasernejad, Ayesha Asloob Qureshi
    arXivPreprintCC BY 4.0Full textAlgebra & number theoryarXiv
  3. ASSOCIATED PRIMES, WITNESSES, AND OMEGA INVARIANTS OF MONOMIAL IDEALS
    Jacob Miller, Mehrdad Nasernejad, Ştefan O Tohǎneanu
    HALPreprintCC BY 4.0Full textAlgebra & number theoryHALarXiv
  4. Polynomial configurations and pointwise averages along Piatetski--Shapiro primes
    Hamed Mousavi
    arXivPreprintarXiv licenceFull textAlgebra & number theoryCombinatorics & discrete mathematicsarXiv
  5. Almost primes in short intervals and averages of long Dirichlet polynomial approximations to (ζ'/ζ)k
    Siegfred Baluyot, John Brian Conrey, Ayesha Irfan +1
    arXivPreprintCC BY 4.0Full textAlgebra & number theoryMathematical physicsarXiv

Cite

BibTeXRIS
BibTeX
@misc{trung2026polynomial,
  title = {{Polynomial bounds for the stability of associated primes of monomial ideals}},
  author = {Tran Nam Trung},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.25422},
  eprint = {2610.25422},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.25422},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Polynomial bounds for the stability of associated primes of monomial ideals
AU  - Tran Nam Trung
PY  - 2026
DA  - 2026/10/18
PB  - arXiv
DO  - 10.48550/arxiv.2610.25422
UR  - https://arxiv.org/abs/2610.25422
AB  - Let I be a nonzero proper monomial ideal in k[x_1,…,x_r], and let d be the maximal degree of a minimal monomial generator. We prove that the index of stability of the associated primes of its ordinary powers satisfies astab(I)≤((4e)^r−1)/4 d^r−1 (r≥3), while astab(I)=1 for r=2. If I has s≥2 minimal generators, we also obtain astab(I)≤min{r,s}(s−1)d^min{r−1,s−2}. The first estimate answers Hoa's question on polynomial bounds. The proofs reduce stabilization to integer optimization for a matrix whose last row records the power of the ideal. We bound the degrees of its circuits and Graver basis elements and prove proximity estimates for optimal solutions in fixed fibers. A Steinitz rearrangement and a lattice-point count in the difference body of a simplex give the generator-independent bound. The matrix estimates hold for arbitrary indexed integral configurations in a simplex and real linear objective functions. The stability bounds hold over an arbitrary field, without normality or equigeneration assumptions.
ER  -