{"paper":{"title":"Slice rank and partition rank of the determinant","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CC"],"primary_cat":"math.CO","authors_text":"Amichai Lampert, Guy Moshkovitz","submitted_at":"2025-09-08T02:35:50Z","abstract_excerpt":"The Laplace expansion expresses the $n \\times n$ determinant $\\det_n$ as a sum of $n$ products. Do shorter expansions exist? In this paper we:\n  - Fully determine the slice rank decompositions of $\\det_n$ (where each product must contain a linear factor): In this case, we show that $n$ summands are necessary, and moreover, the only such expansions with $n$ summands are equivalent (in a precise sense) to the Laplace expansion. \n  - Prove a logarithmic lower bound for the partition rank of $\\det_n$ (where each product is of multilinear forms): In this case, we show that at least $\\log_2(n)+1$ su"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2509.06294","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/2509.06294/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"}