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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1401.4309","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2014-01-17T11:15:31Z","cross_cats_sorted":[],"title_canon_sha256":"601a17a3ac0583eb84c56be603180b689f06e48c2cc86d6e712d9542b6978a4e","abstract_canon_sha256":"d4e71d9417a14c0a91e4cb50927fd36af0156e7da3c2205d7ffbc85a33444410"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:42:06.025636Z","signature_b64":"zceiTtMw3Ku0KSvJ/siCBHeDgeri4+vn+1xSEAnNpcV+GbU2E9BcLJBglmUbJAje1PMdKrXFCw2LrfLLxdmQDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"41bdfbd8a395aaa6e87657577226973088d581b2436cb88efac73f9871a88e4d","last_reissued_at":"2026-05-18T02:42:06.025185Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:42:06.025185Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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. 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