{"paper":{"title":"Bounding the socles of powers of squarefree monomial ideals","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"J\\\"urgen Herzog, Takayuki Hibi","submitted_at":"2013-08-25T11:40:03Z","abstract_excerpt":"Let $S=K[x_1,\\ldots,x_n]$ be the polynomial ring in $n$ variables over a field $K$ and $I\\subset S$ a squarefree monomial ideal. In the present paper we are interested in the monomials $u \\in S$ belonging to the socle $\\Soc(S/I^{k})$ of $S/I^{k}$, i.e., $u \\not\\in I^{k}$ and $ux_{i} \\in I^{k}$ for $1 \\leq i \\leq n$. We prove that if a monomial $x_1^{a_1}\\cdots x_n^{a_n}$ belongs to $\\Soc(S/I^{k})$, then $a_i\\leq k-1$ for all $1 \\leq i \\leq n$. We then discuss squarefree monomial ideals $I \\subset S$ for which $x_{[n]}^{k-1} \\in \\Soc(S/I^{k})$, where $x_{[n]} = x_{1}x_{2}\\cdots x_{n}$. Furtherm"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1308.5400","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}