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It is well-known that the degree complexity with respect to the graded reverse lexicographic order is equal to the Castelnuovo-Mumford regularity (\\cite{BS}). However, little is known about the degree complexity with respect to the graded lexicographic order (\\cite{A}, \\cite{CS}).\n  In this paper, we study the degree complexity of a smooth irreducible surface in $\\p^4$ with respect to the graded lexicographic order and its geometric mean"},"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":"1008.0978","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2010-08-05T14:06:33Z","cross_cats_sorted":["math.AC"],"title_canon_sha256":"e83f37cf0d7c623c502802a1d470eec5ad99740fd63dc8988de0a0ceda144a5d","abstract_canon_sha256":"894ec1f8fda040d5940c91fd805eba6ed9b871b2e39762eded72cb0be1230cd8"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:25:11.294037Z","signature_b64":"7htQnituasbewkVTmE8EoM14cMZyAJmSN064JEDNhAgEuwG+vQ0y4GggBHwfnbwM4WjT2Neg969wBsLtsuLFAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"429439dc2e59b42caefe7bf34617ee5b737f6dc4eec92d4a4b644dde385ce17f","last_reissued_at":"2026-05-18T04:25:11.293481Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:25:11.293481Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The Degree Complexity of Smooth Surfaces of codimension 2","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AC"],"primary_cat":"math.AG","authors_text":"Jeaman Ahn, Sijong Kwak, YeongSeok Song","submitted_at":"2010-08-05T14:06:33Z","abstract_excerpt":"D.Bayer and D.Mumford introduced the degree complexity of a projective scheme for the given term order as the maximal degree of the reduced Gr\\\"{o}bner basis. 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