{"paper":{"title":"On the depth and Stanley depth of integral closure of powers of monomial ideals","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.AC","authors_text":"S. A. Seyed Fakhari","submitted_at":"2018-08-09T15:02:03Z","abstract_excerpt":"Let $\\mathbb{K}$ be a field and $S=\\mathbb{K}[x_1,\\dots,x_n]$ be the polynomial ring in $n$ variables over $\\mathbb{K}$. Assume that $G$ is a graph with edge ideal $I(G)$. We prove that the modules $S/\\overline{I(G)^k}$ and $\\overline{I(G)^k}/\\overline{I(G)^{k+1}}$ satisfy Stanley's inequality for every integer $k\\gg 0$. If $G$ is a non-bipartite graph, we show that the ideals $\\overline{I(G)^k}$ satisfy Stanley's inequality for all $k\\gg 0$. For every connected bipartite graph $G$ (with at least one edge), we prove that ${\\rm sdepth}(I(G)^k)\\geq 2$, for any positive integer $k\\leq {\\rm girth}"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1808.03189","kind":"arxiv","version":2},"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"}