On the Stanley length of monomial ideals
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866915920858841088 |
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| author | Cimpoeas, Mircea |
| author_facet | Cimpoeas, Mircea |
| contents | Let $S=K[x_1,\ldots,x_n]$ be the ring of polynomials in $n$ variables over an arbitrary field $K$. Given a finitely generated multigraded module $M$, its Stanley length, denoted by $\operatorname{slength}(M)$, is the minimal length of a Stanley decomposition of $M$. Let $I\subset S$ be a monomial ideal, minimally generated by $m$ monomials. We give an upper bound for $\operatorname{slength}(I)$, in terms of its minimal monomial generators. Also, we give precise formulas for $\operatorname{slength}(I)$, if $n=2$ or $m=2$. Also, we show that if $I$ has linear quotients, then $\operatorname{slength}(I)=m$, and the converse holds in some special cases. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_17935 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the Stanley length of monomial ideals Cimpoeas, Mircea Commutative Algebra 05E40, 06A17, 13A15, 13C15, 13P10 Let $S=K[x_1,\ldots,x_n]$ be the ring of polynomials in $n$ variables over an arbitrary field $K$. Given a finitely generated multigraded module $M$, its Stanley length, denoted by $\operatorname{slength}(M)$, is the minimal length of a Stanley decomposition of $M$. Let $I\subset S$ be a monomial ideal, minimally generated by $m$ monomials. We give an upper bound for $\operatorname{slength}(I)$, in terms of its minimal monomial generators. Also, we give precise formulas for $\operatorname{slength}(I)$, if $n=2$ or $m=2$. Also, we show that if $I$ has linear quotients, then $\operatorname{slength}(I)=m$, and the converse holds in some special cases. |
| title | On the Stanley length of monomial ideals |
| topic | Commutative Algebra 05E40, 06A17, 13A15, 13C15, 13P10 |
| url | https://arxiv.org/abs/2507.17935 |