{"paper":{"title":"A near-quadratic lower bound on the border determinantal complexity of $\\sum_i x_i^n$ via conormal specialization","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"cs.CC","authors_text":"Karthik Sheshadri","submitted_at":"2026-06-11T17:40:51Z","abstract_excerpt":"The border determinantal complexity $\\dcb(f)$ of a polynomial $f$ is the least $m$ such that $f$ is a limit of determinants of $m\\times m$ matrices of affine-linear forms. We prove that for every $n\\ge3$, over $\\CC$, \\[\n  \\dcb\\Big(\\sum_{i=1}^n x_i^n\\Big)\\ \\ge\\ \\frac{(n-1)^2}{4e},\n  \\qquad\n  \\sdcb\\Big(\\sum_{i=1}^n x_i^n\\Big)\\ \\ge\\ \\frac{(n-1)^2}{2e} \\] in the ordinary and symmetric models respectively; both match the known $O(n^2)$ upper bounds up to the constant. To our knowledge these are the first border determinantal lower bounds for an explicit family that are superlinear in the number of "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.13628","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/2606.13628/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"}