{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2015:LD3RN4EMJKHV33Q7HHOSXJY6NA","short_pith_number":"pith:LD3RN4EM","schema_version":"1.0","canonical_sha256":"58f716f08c4a8f5dee1f39dd2ba71e680fc453afc78830cbb73170f3c468fbcf","source":{"kind":"arxiv","id":"1501.06046","version":4},"attestation_state":"computed","paper":{"title":"Rational maps $H$ for which $K(tH)$ has transcendence degree 2 over $K$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Michiel de Bondt","submitted_at":"2015-01-24T14:24:39Z","abstract_excerpt":"We classify all rational maps $H \\in K(x)^n$ for which ${\\rm trdeg}_K K(tH_1,tH_2,\\ldots,tH_n) \\le 2$, where $K$ is any field and $t$ is another indeterminate.\n  Furthermore, we classify all such maps for which additionally $JH \\cdot H = {\\rm tr} JH \\cdot H$ (where $JH$ is the Jacobian matrix of $H$), i.e. $$ \\sum_{i=1}^n H_i \\frac{\\partial}{\\partial x_i} H_k = \\sum_{i=1}^n H_k \\frac{\\partial}{\\partial x_i} H_i $$ for all $k \\le n$. This generalizes a theorem of Paul Gordan and Max N\\\"other, in which both sides and the characteristic of $K$ are assumed to be zero.\n  Besides this, we use some o"},"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":"1501.06046","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2015-01-24T14:24:39Z","cross_cats_sorted":[],"title_canon_sha256":"064e8f596d6b3556a4231dfd05d1a0882eb32069847addc80e9ab151e522986d","abstract_canon_sha256":"6138e6819f309f758f74d997baab395ae9c4f42dce3421c3a6667788b1dcb52b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:31:29.173244Z","signature_b64":"nZ/UBBNWzoIfwN8oLPiTwARqRIf1ryW0UXFjPte8wqxTADwnSO66gLK80CP+x5dMb2YjH5E8z07v3EvbYrSjAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"58f716f08c4a8f5dee1f39dd2ba71e680fc453afc78830cbb73170f3c468fbcf","last_reissued_at":"2026-05-18T00:31:29.172729Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:31:29.172729Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Rational maps $H$ for which $K(tH)$ has transcendence degree 2 over $K$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Michiel de Bondt","submitted_at":"2015-01-24T14:24:39Z","abstract_excerpt":"We classify all rational maps $H \\in K(x)^n$ for which ${\\rm trdeg}_K K(tH_1,tH_2,\\ldots,tH_n) \\le 2$, where $K$ is any field and $t$ is another indeterminate.\n  Furthermore, we classify all such maps for which additionally $JH \\cdot H = {\\rm tr} JH \\cdot H$ (where $JH$ is the Jacobian matrix of $H$), i.e. $$ \\sum_{i=1}^n H_i \\frac{\\partial}{\\partial x_i} H_k = \\sum_{i=1}^n H_k \\frac{\\partial}{\\partial x_i} H_i $$ for all $k \\le n$. This generalizes a theorem of Paul Gordan and Max N\\\"other, in which both sides and the characteristic of $K$ are assumed to be zero.\n  Besides this, we use some o"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1501.06046","kind":"arxiv","version":4},"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":"1501.06046","created_at":"2026-05-18T00:31:29.172811+00:00"},{"alias_kind":"arxiv_version","alias_value":"1501.06046v4","created_at":"2026-05-18T00:31:29.172811+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1501.06046","created_at":"2026-05-18T00:31:29.172811+00:00"},{"alias_kind":"pith_short_12","alias_value":"LD3RN4EMJKHV","created_at":"2026-05-18T12:29:29.992203+00:00"},{"alias_kind":"pith_short_16","alias_value":"LD3RN4EMJKHV33Q7","created_at":"2026-05-18T12:29:29.992203+00:00"},{"alias_kind":"pith_short_8","alias_value":"LD3RN4EM","created_at":"2026-05-18T12:29:29.992203+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/LD3RN4EMJKHV33Q7HHOSXJY6NA","json":"https://pith.science/pith/LD3RN4EMJKHV33Q7HHOSXJY6NA.json","graph_json":"https://pith.science/api/pith-number/LD3RN4EMJKHV33Q7HHOSXJY6NA/graph.json","events_json":"https://pith.science/api/pith-number/LD3RN4EMJKHV33Q7HHOSXJY6NA/events.json","paper":"https://pith.science/paper/LD3RN4EM"},"agent_actions":{"view_html":"https://pith.science/pith/LD3RN4EMJKHV33Q7HHOSXJY6NA","download_json":"https://pith.science/pith/LD3RN4EMJKHV33Q7HHOSXJY6NA.json","view_paper":"https://pith.science/paper/LD3RN4EM","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1501.06046&json=true","fetch_graph":"https://pith.science/api/pith-number/LD3RN4EMJKHV33Q7HHOSXJY6NA/graph.json","fetch_events":"https://pith.science/api/pith-number/LD3RN4EMJKHV33Q7HHOSXJY6NA/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/LD3RN4EMJKHV33Q7HHOSXJY6NA/action/timestamp_anchor","attest_storage":"https://pith.science/pith/LD3RN4EMJKHV33Q7HHOSXJY6NA/action/storage_attestation","attest_author":"https://pith.science/pith/LD3RN4EMJKHV33Q7HHOSXJY6NA/action/author_attestation","sign_citation":"https://pith.science/pith/LD3RN4EMJKHV33Q7HHOSXJY6NA/action/citation_signature","submit_replication":"https://pith.science/pith/LD3RN4EMJKHV33Q7HHOSXJY6NA/action/replication_record"}},"created_at":"2026-05-18T00:31:29.172811+00:00","updated_at":"2026-05-18T00:31:29.172811+00:00"}