{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:QWDRB66T7ZN6ZLPQHKYY2FN7DP","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"c6352b5e3df4c5d53b307ce9a4c74714b12fad4cd797a05acaeef3029f5aecf2","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2017-08-21T15:32:44Z","title_canon_sha256":"0a542565b46cba6e21f2991e09621d0f4f362a8a5dc34ec6e9342ca5377d5157"},"schema_version":"1.0","source":{"id":"1708.06287","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1708.06287","created_at":"2026-05-18T00:32:57Z"},{"alias_kind":"arxiv_version","alias_value":"1708.06287v2","created_at":"2026-05-18T00:32:57Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1708.06287","created_at":"2026-05-18T00:32:57Z"},{"alias_kind":"pith_short_12","alias_value":"QWDRB66T7ZN6","created_at":"2026-05-18T12:31:39Z"},{"alias_kind":"pith_short_16","alias_value":"QWDRB66T7ZN6ZLPQ","created_at":"2026-05-18T12:31:39Z"},{"alias_kind":"pith_short_8","alias_value":"QWDRB66T","created_at":"2026-05-18T12:31:39Z"}],"graph_snapshots":[{"event_id":"sha256:f399e6e6b39ad4086eac7cfa49c3905f091cc07993c722a2b62870eb497cdbea","target":"graph","created_at":"2026-05-18T00:32:57Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"This article generalizes joint work of the first author and I. Swanson to the $s$-multiplicity recently introduced by the second author. For $k$ a field and $X = [ x_{i,j}]$ a $m \\times n$-matrix of variables, we utilize Gr\\\"obner bases to give a closed form the length $\\lambda( k[X] / (I_2(X) + \\mathfrak{m}^{ \\lceil sq \\rceil} + \\mathfrak{m}^{[q]} ))$ where $s \\in \\mathbf{Z}[p^{-1}]$, $q$ is a sufficiently large power of $p$, and $\\mathfrak{m}$ is the homogeneous maximal ideal of $k[X]$. This shows this length is always eventually a {\\it polynomial} function of $q$ for all $s$.","authors_text":"Lance Edward Miller, William D. Taylor","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2017-08-21T15:32:44Z","title":"The s-multiplicity function of 2x2-determinantal rings"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1708.06287","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:7ead98e2fa395a39e4428c58671987c7a49ebff27059902517c743321f96046a","target":"record","created_at":"2026-05-18T00:32:57Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"c6352b5e3df4c5d53b307ce9a4c74714b12fad4cd797a05acaeef3029f5aecf2","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2017-08-21T15:32:44Z","title_canon_sha256":"0a542565b46cba6e21f2991e09621d0f4f362a8a5dc34ec6e9342ca5377d5157"},"schema_version":"1.0","source":{"id":"1708.06287","kind":"arxiv","version":2}},"canonical_sha256":"858710fbd3fe5becadf03ab18d15bf1be7aaaeb12fc5635c4efcc8b2cf8c99e4","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"858710fbd3fe5becadf03ab18d15bf1be7aaaeb12fc5635c4efcc8b2cf8c99e4","first_computed_at":"2026-05-18T00:32:57.403929Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:32:57.403929Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"SKue55T+a4ZZl1ryxwpg1zqdHT5QiFKFlQky+/GrsYTFreSV3yMmsgv2X3kE6GyWhHc1P1jEIg8mPjNq3IjgBw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:32:57.404520Z","signed_message":"canonical_sha256_bytes"},"source_id":"1708.06287","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:7ead98e2fa395a39e4428c58671987c7a49ebff27059902517c743321f96046a","sha256:f399e6e6b39ad4086eac7cfa49c3905f091cc07993c722a2b62870eb497cdbea"],"state_sha256":"ed6ba58d98ddf47b1fe40ee6dbad6225661ddd8738c61c85d39c8a0df91b6bee"}