{"paper":{"title":"A Lower Bound on Determinantal Complexity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG"],"primary_cat":"cs.CC","authors_text":"Ben lee Volk, Mrinal Kumar","submitted_at":"2020-09-05T03:50:35Z","abstract_excerpt":"The determinantal complexity of a polynomial $P \\in \\mathbb{F}[x_1, \\ldots, x_n]$ over a field $\\mathbb{F}$ is the dimension of the smallest matrix $M$ whose entries are affine functions in $\\mathbb{F}[x_1, \\ldots, x_n]$ such that $P = Det(M)$. We prove that the determinantal complexity of the polynomial $\\sum_{i = 1}^n x_i^n$ is at least $1.5n - 3$.\n  For every $n$-variate polynomial of degree $d$, the determinantal complexity is trivially at least $d$, and it is a long standing open problem to prove a lower bound which is super linear in $\\max\\{n,d\\}$. Our result is the first lower bound for"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2009.02452","kind":"arxiv","version":2},"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/2009.02452/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"}