{"paper":{"title":"The behavior of Stanley depth under polarization","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Bogdan Ichim, Julio Jos\\'e Moyano-Fern\\'andez, Lukas Katth\\\"an","submitted_at":"2014-01-17T11:15:31Z","abstract_excerpt":"Let $K$ be a field, $R=K[X_1, ..., X_n]$ be the polynomial ring and $J \\subsetneq I$ two monomial ideals in $R$. In this paper we show that $\\mathrm{sdepth}\\ {I/J} - \\mathrm{depth}\\ {I/J} = \\mathrm{sdepth}\\ {I^p/J^p}-\\mathrm{depth}\\ {I^p/J^p}$, where $\\mathrm{sdepth}\\ I/J$ denotes the Stanley depth and $I^p$ denotes the polarization. This solves a conjecture by Herzog and reduces the famous Stanley conjecture (for modules of the form $I/J$) to the squarefree case. As a consequence, the Stanley conjecture for algebras of the form $R/I$ and the well-known combinatorial conjecture that every Cohe"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1401.4309","kind":"arxiv","version":3},"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"}