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We show that there exist $\\alpha_{1},\\alpha_{2}\\in\\mathbb{R}, \\beta\\in\\frac{1}{2}\\mathbb{Z}$, and $l\\in\\mathbb{N}$ such that $\\alpha_{1}n^{\\beta}l^{n}\\leq c_{n}^{G}(W)\\leq\\alpha_{2}n^{\\beta}l^{n}$. Furthermore, if W has a unit then the asymptotic behavior of $c_{n}^{G}(W)$ is $\\alpha n^{\\beta}l^{n}$ where $\\alpha\\in\\mathbb{R}, \\beta\\in\\frac{1}{2}\\mathbb{Z}, l\\in\\mathbb{N}$."},"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":"1504.00373","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2015-04-01T20:07:19Z","cross_cats_sorted":[],"title_canon_sha256":"bb56c67ca0e6f8b3ae2b2629560dabb6956e7ff2f2e41cd1ff24da9b9c7417c9","abstract_canon_sha256":"11cf33f1159c059f11df6db6e36858a42c9931b2d34f00d60a7c458932db8520"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:19:43.445542Z","signature_b64":"iMD5PNywMZ4fCvjGLlU3fL6pzbt/cA6L8EeDdtVXMhY0RkuDjRftZszDzpPPy16v9V+OGsaSjEtR1HqvZ8RfCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"505fa07eb71b36a1b4e8dfacaaa098e106153b468e295fbc75f1516b350f7cfe","last_reissued_at":"2026-05-18T02:19:43.445080Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:19:43.445080Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The Asymptotic Behavior of the Codimension Sequence of Affine G - Graded Algebras","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RA","authors_text":"Yuval Shpigelman","submitted_at":"2015-04-01T20:07:19Z","abstract_excerpt":"Let W be an affine PI algebra over a field of characteristic zero graded by a finite group G. 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Furthermore, if W has a unit then the asymptotic behavior of $c_{n}^{G}(W)$ is $\\alpha n^{\\beta}l^{n}$ where $\\alpha\\in\\mathbb{R}, \\beta\\in\\frac{1}{2}\\mathbb{Z}, l\\in\\mathbb{N}$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1504.00373","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1504.00373","created_at":"2026-05-18T02:19:43.445143+00:00"},{"alias_kind":"arxiv_version","alias_value":"1504.00373v1","created_at":"2026-05-18T02:19:43.445143+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1504.00373","created_at":"2026-05-18T02:19:43.445143+00:00"},{"alias_kind":"pith_short_12","alias_value":"KBP2A7VXDM3K","created_at":"2026-05-18T12:29:27.538025+00:00"},{"alias_kind":"pith_short_16","alias_value":"KBP2A7VXDM3KDNHI","created_at":"2026-05-18T12:29:27.538025+00:00"},{"alias_kind":"pith_short_8","alias_value":"KBP2A7VX","created_at":"2026-05-18T12:29:27.538025+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/KBP2A7VXDM3KDNHI36WKVIEY4E","json":"https://pith.science/pith/KBP2A7VXDM3KDNHI36WKVIEY4E.json","graph_json":"https://pith.science/api/pith-number/KBP2A7VXDM3KDNHI36WKVIEY4E/graph.json","events_json":"https://pith.science/api/pith-number/KBP2A7VXDM3KDNHI36WKVIEY4E/events.json","paper":"https://pith.science/paper/KBP2A7VX"},"agent_actions":{"view_html":"https://pith.science/pith/KBP2A7VXDM3KDNHI36WKVIEY4E","download_json":"https://pith.science/pith/KBP2A7VXDM3KDNHI36WKVIEY4E.json","view_paper":"https://pith.science/paper/KBP2A7VX","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1504.00373&json=true","fetch_graph":"https://pith.science/api/pith-number/KBP2A7VXDM3KDNHI36WKVIEY4E/graph.json","fetch_events":"https://pith.science/api/pith-number/KBP2A7VXDM3KDNHI36WKVIEY4E/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/KBP2A7VXDM3KDNHI36WKVIEY4E/action/timestamp_anchor","attest_storage":"https://pith.science/pith/KBP2A7VXDM3KDNHI36WKVIEY4E/action/storage_attestation","attest_author":"https://pith.science/pith/KBP2A7VXDM3KDNHI36WKVIEY4E/action/author_attestation","sign_citation":"https://pith.science/pith/KBP2A7VXDM3KDNHI36WKVIEY4E/action/citation_signature","submit_replication":"https://pith.science/pith/KBP2A7VXDM3KDNHI36WKVIEY4E/action/replication_record"}},"created_at":"2026-05-18T02:19:43.445143+00:00","updated_at":"2026-05-18T02:19:43.445143+00:00"}