{"paper":{"title":"Rosenbrock's Theorem characterizes Pr\\\"{u}fer domains","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Vanni Noferini","submitted_at":"2026-05-31T23:32:18Z","abstract_excerpt":"Rosenbrock's Theorem is a result, originally motivated by engineering applications, that was first proved over the univariate polynomial rings $\\mathcal{R} = \\mathbb{R}[x]$ and $\\mathcal{R}=\\mathbb{C}[x]$, and later established to hold for every elementary divisor domain $\\mathcal{R}$. Under some coprimality assumptions on certain submatrices, Rosenbrock's Theorem connects the Smith form of a matrix $P$ over $\\mathcal{R}$ to the Smith-McMillan form of a matrix $G$ over the field of fractions of $\\mathcal{R}$, where $G$ is a Schur complement in $P$. If $\\mathcal{R}$ is not an elementary divisor"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.01497","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2606.01497/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}