{"paper":{"title":"Powers of matrices with all principal minors equal to 1","license":"http://creativecommons.org/publicdomain/zero/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.AC","authors_text":"Darij Grinberg","submitted_at":"2026-06-27T15:27:48Z","abstract_excerpt":"Consider a square matrix $A$ whose all principal minors are equal to $1$. Over a field, this property is inherited by any power of $A$, but this is not the case over an arbitrary commutative ring. We show that it is the case over any regular ring, and also over the ring $\\mathbb{Z} / d$ for any integer $d$, and in some other settings (quotients of Pr\\\"ufer domains and principal quotients of normal domains). This generalizes Problem B5 of the 2021 Putnam contest.\n  Over arbitrary commutative rings, we identify a stronger property that is always inherited by powers: We say that a matrix $A = \\le"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.28976","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/2606.28976/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"}