{"paper":{"title":"On the Limits of Depth Reduction at Depth 3 Over Small Finite Fields","license":"http://creativecommons.org/licenses/by-nc-sa/3.0/","headline":"","cross_cats":[],"primary_cat":"cs.CC","authors_text":"Partha Mukhopadhyay, Suryajith Chillara","submitted_at":"2013-12-31T17:08:36Z","abstract_excerpt":"Recently, Gupta et.al. [GKKS2013] proved that over Q any $n^{O(1)}$-variate and $n$-degree polynomial in VP can also be computed by a depth three $\\Sigma\\Pi\\Sigma$ circuit of size $2^{O(\\sqrt{n}\\log^{3/2}n)}$. Over fixed-size finite fields, Grigoriev and Karpinski proved that any $\\Sigma\\Pi\\Sigma$ circuit that computes $Det_n$ (or $Perm_n$) must be of size $2^{\\Omega(n)}$ [GK1998]. In this paper, we prove that over fixed-size finite fields, any $\\Sigma\\Pi\\Sigma$ circuit for computing the iterated matrix multiplication polynomial of $n$ generic matrices of size $n\\times n$, must be of size $2^{"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1401.0189","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":""},"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"}