{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2018:TEZXY33KRJBKFBVTSTSKA5ZNEX","short_pith_number":"pith:TEZXY33K","schema_version":"1.0","canonical_sha256":"99337c6f6a8a42a286b394e4a0772d25e1c7977d5a93513d71a1d6b16f00a589","source":{"kind":"arxiv","id":"1801.06977","version":1},"attestation_state":"computed","paper":{"title":"Density function for the second coefficient of the Hilbert-Kunz function","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG"],"primary_cat":"math.AC","authors_text":"Mandira Mondal, Vijaylaxmi Trivedi","submitted_at":"2018-01-22T07:07:05Z","abstract_excerpt":"We prove that, analogous to the HK density function, (used for studying the Hilbert-Kunz multiplicity, the leading coefficient of the HK function), there exists a $\\beta$-density function $g_{R, {\\bf m}}:[0,\\infty)\\longrightarrow {\\mathbb R}$, where $(R, {\\bf m})$ is the homogeneous coordinate ring associated to the toric pair $(X, D)$, such that $$\\int_0^{\\infty}g_{R, {\\bf m}}(x)dx = \\beta(R, {\\bf m}),$$ where $\\beta(R, {\\bf m})$ is the second coefficient of the Hilbert-Kunz function for $(R, {\\bf m})$, as constructed by Huneke-McDermott-Monsky.\n  Moreover we prove, (1) the function $g_{R, {\\"},"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":"1801.06977","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2018-01-22T07:07:05Z","cross_cats_sorted":["math.AG"],"title_canon_sha256":"94608e3fdcd18d115320622c6a3e1473c732e692d4b8587bfc2385238f468f6f","abstract_canon_sha256":"08eb9359032fd972ccbf096345ca7929a14e7f8c4bbe8a232f1ec5c3e4932392"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:25:22.610761Z","signature_b64":"j6IiiDmkVtAg2FEPFil3+juaUn8kTjevhzl9G4tW7zk9f1VPuZhon6PiYpeZbcSWLA08CxXRZi4pKAGtAyOeCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"99337c6f6a8a42a286b394e4a0772d25e1c7977d5a93513d71a1d6b16f00a589","last_reissued_at":"2026-05-18T00:25:22.610099Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:25:22.610099Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Density function for the second coefficient of the Hilbert-Kunz function","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG"],"primary_cat":"math.AC","authors_text":"Mandira Mondal, Vijaylaxmi Trivedi","submitted_at":"2018-01-22T07:07:05Z","abstract_excerpt":"We prove that, analogous to the HK density function, (used for studying the Hilbert-Kunz multiplicity, the leading coefficient of the HK function), there exists a $\\beta$-density function $g_{R, {\\bf m}}:[0,\\infty)\\longrightarrow {\\mathbb R}$, where $(R, {\\bf m})$ is the homogeneous coordinate ring associated to the toric pair $(X, D)$, such that $$\\int_0^{\\infty}g_{R, {\\bf m}}(x)dx = \\beta(R, {\\bf m}),$$ where $\\beta(R, {\\bf m})$ is the second coefficient of the Hilbert-Kunz function for $(R, {\\bf m})$, as constructed by Huneke-McDermott-Monsky.\n  Moreover we prove, (1) the function $g_{R, {\\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1801.06977","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":"1801.06977","created_at":"2026-05-18T00:25:22.610209+00:00"},{"alias_kind":"arxiv_version","alias_value":"1801.06977v1","created_at":"2026-05-18T00:25:22.610209+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1801.06977","created_at":"2026-05-18T00:25:22.610209+00:00"},{"alias_kind":"pith_short_12","alias_value":"TEZXY33KRJBK","created_at":"2026-05-18T12:32:53.628368+00:00"},{"alias_kind":"pith_short_16","alias_value":"TEZXY33KRJBKFBVT","created_at":"2026-05-18T12:32:53.628368+00:00"},{"alias_kind":"pith_short_8","alias_value":"TEZXY33K","created_at":"2026-05-18T12:32:53.628368+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/TEZXY33KRJBKFBVTSTSKA5ZNEX","json":"https://pith.science/pith/TEZXY33KRJBKFBVTSTSKA5ZNEX.json","graph_json":"https://pith.science/api/pith-number/TEZXY33KRJBKFBVTSTSKA5ZNEX/graph.json","events_json":"https://pith.science/api/pith-number/TEZXY33KRJBKFBVTSTSKA5ZNEX/events.json","paper":"https://pith.science/paper/TEZXY33K"},"agent_actions":{"view_html":"https://pith.science/pith/TEZXY33KRJBKFBVTSTSKA5ZNEX","download_json":"https://pith.science/pith/TEZXY33KRJBKFBVTSTSKA5ZNEX.json","view_paper":"https://pith.science/paper/TEZXY33K","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1801.06977&json=true","fetch_graph":"https://pith.science/api/pith-number/TEZXY33KRJBKFBVTSTSKA5ZNEX/graph.json","fetch_events":"https://pith.science/api/pith-number/TEZXY33KRJBKFBVTSTSKA5ZNEX/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/TEZXY33KRJBKFBVTSTSKA5ZNEX/action/timestamp_anchor","attest_storage":"https://pith.science/pith/TEZXY33KRJBKFBVTSTSKA5ZNEX/action/storage_attestation","attest_author":"https://pith.science/pith/TEZXY33KRJBKFBVTSTSKA5ZNEX/action/author_attestation","sign_citation":"https://pith.science/pith/TEZXY33KRJBKFBVTSTSKA5ZNEX/action/citation_signature","submit_replication":"https://pith.science/pith/TEZXY33KRJBKFBVTSTSKA5ZNEX/action/replication_record"}},"created_at":"2026-05-18T00:25:22.610209+00:00","updated_at":"2026-05-18T00:25:22.610209+00:00"}