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Using this result, Gupta, Kamath, Kayal and Saptharishi gave a $\\exp(O(\\sqrt{d\\log(d)\\log(n)\\log(s)}))$ upper bound for the size of the smallest depth three circuit computing a $n$-variate polynomial of degree $d=n^{O(1)}$ given by a circuit of size $s$.\n  We improve here Koiran's bound. Indeed, we show that if we reduce an arithmetic circuit to depth four, then the siz"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1304.5777","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2013-04-21T18:53:23Z","cross_cats_sorted":[],"title_canon_sha256":"f76bacafa91fbe06f69175a81a2c25fe042f52fd1bc235580d00961f78c91569","abstract_canon_sha256":"0e031cdbd96ea03634c2c0313d9cd3fea226a112971f0f8312d6826ec655e6e0"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:51:48.652389Z","signature_b64":"WNKI0H9eSM4laYUDaqw3NIb5FlIi2tiVkcA8NO2SgPsj5L6kDi5LAH17yytZvkhN5xVyPZ6dGsLzGyBFGzNVCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8ce8ce1ab7a8e4ccaad4dfc916b88307302fbbda06cbe262d241e5ff0dbff3e3","last_reissued_at":"2026-05-18T02:51:48.651971Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:51:48.651971Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Improved bounds for reduction to depth 4 and depth 3","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.CC","authors_text":"S\\'ebastien Tavenas","submitted_at":"2013-04-21T18:53:23Z","abstract_excerpt":"Koiran showed that if a $n$-variate polynomial of degree $d$ (with $d=n^{O(1)}$) is computed by a circuit of size $s$, then it is also computed by a homogeneous circuit of depth four and of size $2^{O(\\sqrt{d}\\log(d)\\log(s))}$. 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