{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:I6JP5B7YVFBSEQXXH4ECMHSLH4","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":"48a12fa3d49b47d2c53d96983af72a5151a8947dec1c4920bb9992b6b6d4d31e","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2016-05-20T09:38:57Z","title_canon_sha256":"424a308688a1000f9dac2bc63c9f74906528fea7cf8a12afdd377fc4f5d1facf"},"schema_version":"1.0","source":{"id":"1605.06263","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1605.06263","created_at":"2026-05-18T01:14:19Z"},{"alias_kind":"arxiv_version","alias_value":"1605.06263v1","created_at":"2026-05-18T01:14:19Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1605.06263","created_at":"2026-05-18T01:14:19Z"},{"alias_kind":"pith_short_12","alias_value":"I6JP5B7YVFBS","created_at":"2026-05-18T12:30:22Z"},{"alias_kind":"pith_short_16","alias_value":"I6JP5B7YVFBSEQXX","created_at":"2026-05-18T12:30:22Z"},{"alias_kind":"pith_short_8","alias_value":"I6JP5B7Y","created_at":"2026-05-18T12:30:22Z"}],"graph_snapshots":[{"event_id":"sha256:10902413d86d2269985b10a3b0f93c544cea4ff624cd450c7274d9abfb079ccc","target":"graph","created_at":"2026-05-18T01:14:19Z","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":"Assume that $K$ is a field and $I_{1}\\subsetneq ...\\subsetneq I_{t}$ is an ascending chain (of length $t$) of ideals in the polynomial ring $K[x_{1},,...,x_{m}]$, for some $m\\geq 1$. Suppose that $I_{j}$ is generated by polynomials of degrees less or equal to some natural number $f(j)\\geq 1$, for any $j=1,...,t$. In the paper we construct, in an elementary way, a natural number $\\mathcal{B}(m,f)$ (depending on $m$ and the function $f$) such that $t\\leq\\mathcal{B}(m,f)$. We also discuss some possible applications of this result.","authors_text":"Grzegorz Pastuszak","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2016-05-20T09:38:57Z","title":"On ascending chains of ideals in the polynomial ring"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1605.06263","kind":"arxiv","version":1},"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":"sha256:7fe8157a02c62c230832fe5e8b76b15e6a674f1548aec218b1e29c8a15bbd5ae","target":"record","created_at":"2026-05-18T01:14:19Z","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":"48a12fa3d49b47d2c53d96983af72a5151a8947dec1c4920bb9992b6b6d4d31e","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2016-05-20T09:38:57Z","title_canon_sha256":"424a308688a1000f9dac2bc63c9f74906528fea7cf8a12afdd377fc4f5d1facf"},"schema_version":"1.0","source":{"id":"1605.06263","kind":"arxiv","version":1}},"canonical_sha256":"4792fe87f8a9432242f73f08261e4b3f3a4a40e506b4d2fc38c4a79fcd8be65c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"4792fe87f8a9432242f73f08261e4b3f3a4a40e506b4d2fc38c4a79fcd8be65c","first_computed_at":"2026-05-18T01:14:19.413396Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:14:19.413396Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"bTm4OA8evMQlCKfUmphBJ5zyKd7HsnFSNy8N1wZd1BO3JWMtlkqTd6OfE8CSPqYyUlZ+MCWVbMKY2Ws0k+VoDA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:14:19.413989Z","signed_message":"canonical_sha256_bytes"},"source_id":"1605.06263","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:7fe8157a02c62c230832fe5e8b76b15e6a674f1548aec218b1e29c8a15bbd5ae","sha256:10902413d86d2269985b10a3b0f93c544cea4ff624cd450c7274d9abfb079ccc"],"state_sha256":"802c5ed98f0133a8f45bd5e3be1f3972d67bc3c83859c4a833ab61f64ff51b61"}