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PreprintCC BY 4.0arXiv

A comparison of the v-number of a monomial ideal and its integral closure

  • Prativa Biswas
  • Mousumi Mandal
  • Partha Phukan
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Abstract

Let I be a monomial ideal in a standard graded polynomial ring and let I̅ denote its integral closure. We study the relationship between v(I) and v(I̅). We prove that v(I̅) ≤ v(I) for monomial ideals in two variables, for equigenerated monomial ideals in three variables and for several special classes of monomial ideals, while providing examples showing that this inequality does not hold in general. For the edge ideal I(G) of a connected graph G, we show that v(I(G)k)=v(I(G)k̅) = 2k−1 for all k ≥ 1+|E(G)|. Moreover, when G is disconnected, we prove that v(I(G)k̅)≤v(I(G)k) for all sufficiently large k.

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Cite

BibTeXRIS
BibTeX
@misc{biswas2026comparison,
  title = {{A comparison of the v-number of a monomial ideal and its integral closure}},
  author = {Prativa Biswas and Mousumi Mandal and Partha Phukan},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.05044},
  eprint = {2609.05044},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.05044},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - A comparison of the v-number of a monomial ideal and its integral closure
AU  - Prativa Biswas
AU  - Mousumi Mandal
AU  - Partha Phukan
PY  - 2026
DA  - 2026/09/04
PB  - arXiv
DO  - 10.48550/arxiv.2609.05044
UR  - https://arxiv.org/abs/2609.05044
AB  - Let I be a monomial ideal in a standard graded polynomial ring and let I̅ denote its integral closure. We study the relationship between v(I) and v(I̅). We prove that v(I̅) ≤ v(I) for monomial ideals in two variables, for equigenerated monomial ideals in three variables and for several special classes of monomial ideals, while providing examples showing that this inequality does not hold in general. For the edge ideal I(G) of a connected graph G, we show that v(I(G)^k)=v(I(G)^k̅) = 2k−1 for all k ≥ 1+|E(G)|. Moreover, when G is disconnected, we prove that v(I(G)^k̅)≤v(I(G)^k) for all sufficiently large k.
ER  -