{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:QXK2JEV6QXVFAGV25ZR67GWKR7","short_pith_number":"pith:QXK2JEV6","canonical_record":{"source":{"id":"1008.0156","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2010-08-01T07:53:55Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"d0cb95c65631e14277b64ebd7f54d2daf152be7bd75b275f8f43fae1cbdaba2a","abstract_canon_sha256":"d84c4b1715a591d72454229c1b20041d840c1c61a2ea39148b8cc09d64acf572"},"schema_version":"1.0"},"canonical_sha256":"85d5a492be85ea501abaee63ef9aca8fe9e152548fe7c73d4e7dae57fa85f248","source":{"kind":"arxiv","id":"1008.0156","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1008.0156","created_at":"2026-05-18T04:18:46Z"},{"alias_kind":"arxiv_version","alias_value":"1008.0156v1","created_at":"2026-05-18T04:18:46Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1008.0156","created_at":"2026-05-18T04:18:46Z"},{"alias_kind":"pith_short_12","alias_value":"QXK2JEV6QXVF","created_at":"2026-05-18T12:26:13Z"},{"alias_kind":"pith_short_16","alias_value":"QXK2JEV6QXVFAGV2","created_at":"2026-05-18T12:26:13Z"},{"alias_kind":"pith_short_8","alias_value":"QXK2JEV6","created_at":"2026-05-18T12:26:13Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:QXK2JEV6QXVFAGV25ZR67GWKR7","target":"record","payload":{"canonical_record":{"source":{"id":"1008.0156","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2010-08-01T07:53:55Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"d0cb95c65631e14277b64ebd7f54d2daf152be7bd75b275f8f43fae1cbdaba2a","abstract_canon_sha256":"d84c4b1715a591d72454229c1b20041d840c1c61a2ea39148b8cc09d64acf572"},"schema_version":"1.0"},"canonical_sha256":"85d5a492be85ea501abaee63ef9aca8fe9e152548fe7c73d4e7dae57fa85f248","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:18:46.221419Z","signature_b64":"PrWtMSBbnVI4fuvAFyXCXAdtRkWh7oin1mnajuZbVBLZwD11Gp4JKb3Wqjy2s6YYnqGswfcOc0Ci8/HcZPBxDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"85d5a492be85ea501abaee63ef9aca8fe9e152548fe7c73d4e7dae57fa85f248","last_reissued_at":"2026-05-18T04:18:46.220990Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:18:46.220990Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1008.0156","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:18:46Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"do6PVxJ3vMCm5tifziLUfOj6667n7OwTG7FAKcsVKwoZVvEKfnnTtGMvIzVRkaFjV5iOnY0dUPm4ibWGXU5bBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-24T12:56:14.634822Z"},"content_sha256":"64c3f2c1c126796e0bbd57c91e08d7950a77ae020d028c2aaa422ab3957dcada","schema_version":"1.0","event_id":"sha256:64c3f2c1c126796e0bbd57c91e08d7950a77ae020d028c2aaa422ab3957dcada"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:QXK2JEV6QXVFAGV25ZR67GWKR7","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Noether normalizations, reductions of ideals, and matroids","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.AC","authors_text":"Joseph P. Brennan, Neil Epstein","submitted_at":"2010-08-01T07:53:55Z","abstract_excerpt":"We show that given a finitely generated standard-graded algebra of dimension $d$ over an infinite field, its graded Noether normalizations obey a certain kind of `generic exchange', allowing one to pass between any two of them in at most $d$ steps. We prove analogous generic exchange theorems for minimal reductions of an ideal, minimal complete reductions of a set of ideals, and minimal complete reductions of multigraded $k$-algebras. Finally, we unify all these results into a common axiomatic framework by introducing a new topological-combinatorial structure we call a generic matroid, which i"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1008.0156","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:18:46Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"cf9DtnP5BjJdn9Dr3zESfa4esRkhdA1qoxdWbgJLuSJMode5rP5/7zaTnF+ZahhmCXhuOwhBzEiAAHcmWx5FAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-24T12:56:14.635168Z"},"content_sha256":"720722c6234450bcb0975f7dfb7851d93835a56950c9c0f5481ea16f5258a52a","schema_version":"1.0","event_id":"sha256:720722c6234450bcb0975f7dfb7851d93835a56950c9c0f5481ea16f5258a52a"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/QXK2JEV6QXVFAGV25ZR67GWKR7/bundle.json","state_url":"https://pith.science/pith/QXK2JEV6QXVFAGV25ZR67GWKR7/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/QXK2JEV6QXVFAGV25ZR67GWKR7/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-24T12:56:14Z","links":{"resolver":"https://pith.science/pith/QXK2JEV6QXVFAGV25ZR67GWKR7","bundle":"https://pith.science/pith/QXK2JEV6QXVFAGV25ZR67GWKR7/bundle.json","state":"https://pith.science/pith/QXK2JEV6QXVFAGV25ZR67GWKR7/state.json","well_known_bundle":"https://pith.science/.well-known/pith/QXK2JEV6QXVFAGV25ZR67GWKR7/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:QXK2JEV6QXVFAGV25ZR67GWKR7","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":"d84c4b1715a591d72454229c1b20041d840c1c61a2ea39148b8cc09d64acf572","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2010-08-01T07:53:55Z","title_canon_sha256":"d0cb95c65631e14277b64ebd7f54d2daf152be7bd75b275f8f43fae1cbdaba2a"},"schema_version":"1.0","source":{"id":"1008.0156","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1008.0156","created_at":"2026-05-18T04:18:46Z"},{"alias_kind":"arxiv_version","alias_value":"1008.0156v1","created_at":"2026-05-18T04:18:46Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1008.0156","created_at":"2026-05-18T04:18:46Z"},{"alias_kind":"pith_short_12","alias_value":"QXK2JEV6QXVF","created_at":"2026-05-18T12:26:13Z"},{"alias_kind":"pith_short_16","alias_value":"QXK2JEV6QXVFAGV2","created_at":"2026-05-18T12:26:13Z"},{"alias_kind":"pith_short_8","alias_value":"QXK2JEV6","created_at":"2026-05-18T12:26:13Z"}],"graph_snapshots":[{"event_id":"sha256:720722c6234450bcb0975f7dfb7851d93835a56950c9c0f5481ea16f5258a52a","target":"graph","created_at":"2026-05-18T04:18:46Z","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":"We show that given a finitely generated standard-graded algebra of dimension $d$ over an infinite field, its graded Noether normalizations obey a certain kind of `generic exchange', allowing one to pass between any two of them in at most $d$ steps. We prove analogous generic exchange theorems for minimal reductions of an ideal, minimal complete reductions of a set of ideals, and minimal complete reductions of multigraded $k$-algebras. Finally, we unify all these results into a common axiomatic framework by introducing a new topological-combinatorial structure we call a generic matroid, which i","authors_text":"Joseph P. Brennan, Neil Epstein","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2010-08-01T07:53:55Z","title":"Noether normalizations, reductions of ideals, and matroids"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1008.0156","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":"sha256:64c3f2c1c126796e0bbd57c91e08d7950a77ae020d028c2aaa422ab3957dcada","target":"record","created_at":"2026-05-18T04:18:46Z","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":"d84c4b1715a591d72454229c1b20041d840c1c61a2ea39148b8cc09d64acf572","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2010-08-01T07:53:55Z","title_canon_sha256":"d0cb95c65631e14277b64ebd7f54d2daf152be7bd75b275f8f43fae1cbdaba2a"},"schema_version":"1.0","source":{"id":"1008.0156","kind":"arxiv","version":1}},"canonical_sha256":"85d5a492be85ea501abaee63ef9aca8fe9e152548fe7c73d4e7dae57fa85f248","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"85d5a492be85ea501abaee63ef9aca8fe9e152548fe7c73d4e7dae57fa85f248","first_computed_at":"2026-05-18T04:18:46.220990Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:18:46.220990Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"PrWtMSBbnVI4fuvAFyXCXAdtRkWh7oin1mnajuZbVBLZwD11Gp4JKb3Wqjy2s6YYnqGswfcOc0Ci8/HcZPBxDg==","signature_status":"signed_v1","signed_at":"2026-05-18T04:18:46.221419Z","signed_message":"canonical_sha256_bytes"},"source_id":"1008.0156","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:64c3f2c1c126796e0bbd57c91e08d7950a77ae020d028c2aaa422ab3957dcada","sha256:720722c6234450bcb0975f7dfb7851d93835a56950c9c0f5481ea16f5258a52a"],"state_sha256":"4093474b72c77f35dd7cc2681ffbfad2e6d1ddfc42db7fb663719f7fc53ca1ae"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"2z6WDZnY/6Ylk5nTq97d3YoLkHIJ/sZxadCEtKUnmZ+gBdIwxmWhAL/Rd23SeQv8dJ0Dta2KSd7U2/9QCIh5CQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-24T12:56:14.637122Z","bundle_sha256":"8f9a5eb537f4d20433beee92525986eb251db632049c190ec7fc676bc3018699"}}