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PreprintCC BY 4.0HAL

ASSOCIATED PRIMES, WITNESSES, AND OMEGA INVARIANTS OF MONOMIAL IDEALS

  • Jacob Miller
  • Mehrdad Nasernejad
  • Ştefan O Tohǎneanu
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Abstract

We introduce and study the omega invariant of a proper ideal in a Noetherian commutative ring, defined as the number of associated primes of the ideal. Our main objective is to investigate this invariant for monomial ideals and their powers. We characterize associated primes through monomial witnesses and provide an algorithmic procedure for constructing such witnesses from the exponent vectors of the minimal generators. These results lead to explicit formulas and bounds for the omega invariant without requiring the computation of a primary decomposition. We further establish alternative descriptions using irreducible decompositions and Alexander duality. A matrix-based approach is developed to detect associated primes of powers of monomial ideals directly from the exponent matrix of the original ideal. We also investigate the behavior of witnesses under passage from I n to I n+1 and derive corresponding results for edge ideals of graphs.

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Cite

BibTeXRIS
BibTeX
@misc{miller2026associated,
  title = {{ASSOCIATED PRIMES, WITNESSES, AND OMEGA INVARIANTS OF MONOMIAL IDEALS}},
  author = {Jacob Miller and Mehrdad Nasernejad and Ştefan O Tohǎneanu},
  howpublished = {HAL},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.11407},
  eprint = {2609.11407},
  archivePrefix = {arXiv},
  url = {https://hal.science/hal-05745384v1},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - ASSOCIATED PRIMES, WITNESSES, AND OMEGA INVARIANTS OF MONOMIAL IDEALS
AU  - Jacob Miller
AU  - Mehrdad Nasernejad
AU  - Ştefan O Tohǎneanu
PY  - 2026
DA  - 2026/09/10
PB  - HAL
DO  - 10.48550/arxiv.2609.11407
UR  - https://hal.science/hal-05745384v1
AB  - We introduce and study the omega invariant of a proper ideal in a Noetherian commutative ring, defined as the number of associated primes of the ideal. Our main objective is to investigate this invariant for monomial ideals and their powers. We characterize associated primes through monomial witnesses and provide an algorithmic procedure for constructing such witnesses from the exponent vectors of the minimal generators. These results lead to explicit formulas and bounds for the omega invariant without requiring the computation of a primary decomposition. We further establish alternative descriptions using irreducible decompositions and Alexander duality. A matrix-based approach is developed to detect associated primes of powers of monomial ideals directly from the exponent matrix of the original ideal. We also investigate the behavior of witnesses under passage from I n to I n+1 and derive corresponding results for edge ideals of graphs.
ER  -