{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2015:ZV7FPVSV6QHPMJ52F2VK2ZM255","short_pith_number":"pith:ZV7FPVSV","schema_version":"1.0","canonical_sha256":"cd7e57d655f40ef627ba2eaaad659aef4b21d122844c8f1a15b31bd3bcd88c84","source":{"kind":"arxiv","id":"1505.01175","version":3},"attestation_state":"computed","paper":{"title":"Polynomials and harmonic functions on discrete groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG","math.PR"],"primary_cat":"math.GR","authors_text":"Ariel Yadin, Idan Perl, Matthew Tointon, Tom Meyerovitch","submitted_at":"2015-05-05T20:05:58Z","abstract_excerpt":"Alexopoulos proved that on a finitely generated virtually nilpotent group, the restriction of a harmonic function of polynomial growth to a torsion-free nilpotent subgroup of finite index is always a polynomial in the Mal'cev coordinates of that subgroup. For general groups, vanishing of higher-order discrete derivatives gives a natural notion of polynomial maps, which has been considered by Leibman and others. We provide a simple proof of Alexopoulos's result using this notion of polynomials, under the weaker hypothesis that the space of harmonic functions of polynomial growth of degree at mo"},"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":"1505.01175","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2015-05-05T20:05:58Z","cross_cats_sorted":["math.MG","math.PR"],"title_canon_sha256":"fc41b010f440fd4585f9bae48d10e38325f43cbacb0623578b4612be59da25f0","abstract_canon_sha256":"65c26b55e68db9c2513507c9f72bccfd0a898a16f43203455adf716d72a45686"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:16:24.160032Z","signature_b64":"YJS0mOls7dXQ/K+MbXyXYxuzoLRf8+5TWXFkIuwQZNz1X5cUm8wHQxfU5nBH1Lq5XWFWQdjif3LKNqOE5ZrdBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"cd7e57d655f40ef627ba2eaaad659aef4b21d122844c8f1a15b31bd3bcd88c84","last_reissued_at":"2026-05-18T00:16:24.159456Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:16:24.159456Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Polynomials and harmonic functions on discrete groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG","math.PR"],"primary_cat":"math.GR","authors_text":"Ariel Yadin, Idan Perl, Matthew Tointon, Tom Meyerovitch","submitted_at":"2015-05-05T20:05:58Z","abstract_excerpt":"Alexopoulos proved that on a finitely generated virtually nilpotent group, the restriction of a harmonic function of polynomial growth to a torsion-free nilpotent subgroup of finite index is always a polynomial in the Mal'cev coordinates of that subgroup. For general groups, vanishing of higher-order discrete derivatives gives a natural notion of polynomial maps, which has been considered by Leibman and others. We provide a simple proof of Alexopoulos's result using this notion of polynomials, under the weaker hypothesis that the space of harmonic functions of polynomial growth of degree at mo"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1505.01175","kind":"arxiv","version":3},"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":"1505.01175","created_at":"2026-05-18T00:16:24.159565+00:00"},{"alias_kind":"arxiv_version","alias_value":"1505.01175v3","created_at":"2026-05-18T00:16:24.159565+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1505.01175","created_at":"2026-05-18T00:16:24.159565+00:00"},{"alias_kind":"pith_short_12","alias_value":"ZV7FPVSV6QHP","created_at":"2026-05-18T12:29:52.810259+00:00"},{"alias_kind":"pith_short_16","alias_value":"ZV7FPVSV6QHPMJ52","created_at":"2026-05-18T12:29:52.810259+00:00"},{"alias_kind":"pith_short_8","alias_value":"ZV7FPVSV","created_at":"2026-05-18T12:29:52.810259+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/ZV7FPVSV6QHPMJ52F2VK2ZM255","json":"https://pith.science/pith/ZV7FPVSV6QHPMJ52F2VK2ZM255.json","graph_json":"https://pith.science/api/pith-number/ZV7FPVSV6QHPMJ52F2VK2ZM255/graph.json","events_json":"https://pith.science/api/pith-number/ZV7FPVSV6QHPMJ52F2VK2ZM255/events.json","paper":"https://pith.science/paper/ZV7FPVSV"},"agent_actions":{"view_html":"https://pith.science/pith/ZV7FPVSV6QHPMJ52F2VK2ZM255","download_json":"https://pith.science/pith/ZV7FPVSV6QHPMJ52F2VK2ZM255.json","view_paper":"https://pith.science/paper/ZV7FPVSV","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1505.01175&json=true","fetch_graph":"https://pith.science/api/pith-number/ZV7FPVSV6QHPMJ52F2VK2ZM255/graph.json","fetch_events":"https://pith.science/api/pith-number/ZV7FPVSV6QHPMJ52F2VK2ZM255/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/ZV7FPVSV6QHPMJ52F2VK2ZM255/action/timestamp_anchor","attest_storage":"https://pith.science/pith/ZV7FPVSV6QHPMJ52F2VK2ZM255/action/storage_attestation","attest_author":"https://pith.science/pith/ZV7FPVSV6QHPMJ52F2VK2ZM255/action/author_attestation","sign_citation":"https://pith.science/pith/ZV7FPVSV6QHPMJ52F2VK2ZM255/action/citation_signature","submit_replication":"https://pith.science/pith/ZV7FPVSV6QHPMJ52F2VK2ZM255/action/replication_record"}},"created_at":"2026-05-18T00:16:24.159565+00:00","updated_at":"2026-05-18T00:16:24.159565+00:00"}