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In this paper, we prove Lech's conjecture in dimension three when $R$ has equal characteristic. In higher dimension, our method yields substantial partial estimate: $e(R)\\leq (d!/2^d)\\cdot e(S)$ where $d=\\dim R\\geq 4$, in equal characteristic."},"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":"1609.00095","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2016-09-01T03:17:12Z","cross_cats_sorted":[],"title_canon_sha256":"146458be716ab5a040dd122ab80376088bce06078e57b1cc2e2deb9e596c83ef","abstract_canon_sha256":"6934a3365d2f2d2207e52fca8ff648c7110f47bb3b2fb07780908d16b216398b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:23:46.100411Z","signature_b64":"MZa5XjcAicGY4rKWRVt7szHY2H0UJXOdty7gyp1ZeB5lTP2xYhEbjo/8aDZOOHcjoeJr09f5bt/fstF39vZhAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7d6173bfeaab0461a620c1186069a48f35e9511bed7757640dd35668e7aacb5e","last_reissued_at":"2026-05-18T00:23:46.099887Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:23:46.099887Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Lech's conjecture in dimension three","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Linquan Ma","submitted_at":"2016-09-01T03:17:12Z","abstract_excerpt":"Let $(R,m)\\to (S,n)$ be a flat local extension of local rings. 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