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As a special case, we describe all such matrices $M$, which are the Jacobian matrix $J H$ (the matrix of partial derivatives) of a polynomial map $H$ from $K^n$ to $K^m$.\n  Among other things, we show that up to composition with linear maps over $K$, $M = J H$ has only two nonzero columns or only three nonzero rows in this case. In addition, we show that ${\\rm trdeg}_K K(H) = {\\rm rk} J H$ for quadratic poly"},"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":"1601.00579","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2016-01-04T17:41:23Z","cross_cats_sorted":[],"title_canon_sha256":"3139edf54d65f0e1e4113c2ed355d99edede03109f6eeda16ed31a90dcf96a58","abstract_canon_sha256":"cb7be5196796734af5ea202d31e27e0c53667513c8a1277122c8fe8933cb4eb9"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:31:29.087685Z","signature_b64":"PCIPHRkD/PCofc9S5qt0qOuFDBylBxKRqx1vg6WmVPXTxKcXpU4SDMTLGmJ/9FfaCQtFuuVd4A/q8wrJj+DiCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e35731cdcbcc74431ef0e859cd5b7e9c4acf1489744723290103f657e2e0a655","last_reissued_at":"2026-05-18T00:31:29.086888Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:31:29.086888Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Quadratic polynomial maps with Jacobian rank two","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Michiel de Bondt","submitted_at":"2016-01-04T17:41:23Z","abstract_excerpt":"Let $K$ be any field and $x = (x_1,x_2,\\ldots,x_n)$. We classify all matrices $M \\in {\\rm Mat}_{m,n}(K[x])$ whose entries are polynomials of degree at most 1, for which ${\\rm rk} M \\le 2$. As a special case, we describe all such matrices $M$, which are the Jacobian matrix $J H$ (the matrix of partial derivatives) of a polynomial map $H$ from $K^n$ to $K^m$.\n  Among other things, we show that up to composition with linear maps over $K$, $M = J H$ has only two nonzero columns or only three nonzero rows in this case. In addition, we show that ${\\rm trdeg}_K K(H) = {\\rm rk} J H$ for quadratic poly"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1601.00579","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":"1601.00579","created_at":"2026-05-18T00:31:29.087040+00:00"},{"alias_kind":"arxiv_version","alias_value":"1601.00579v4","created_at":"2026-05-18T00:31:29.087040+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1601.00579","created_at":"2026-05-18T00:31:29.087040+00:00"},{"alias_kind":"pith_short_12","alias_value":"4NLTDTOLZR2E","created_at":"2026-05-18T12:29:58.707656+00:00"},{"alias_kind":"pith_short_16","alias_value":"4NLTDTOLZR2EGHXQ","created_at":"2026-05-18T12:29:58.707656+00:00"},{"alias_kind":"pith_short_8","alias_value":"4NLTDTOL","created_at":"2026-05-18T12:29:58.707656+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/4NLTDTOLZR2EGHXQ5BM42W36TR","json":"https://pith.science/pith/4NLTDTOLZR2EGHXQ5BM42W36TR.json","graph_json":"https://pith.science/api/pith-number/4NLTDTOLZR2EGHXQ5BM42W36TR/graph.json","events_json":"https://pith.science/api/pith-number/4NLTDTOLZR2EGHXQ5BM42W36TR/events.json","paper":"https://pith.science/paper/4NLTDTOL"},"agent_actions":{"view_html":"https://pith.science/pith/4NLTDTOLZR2EGHXQ5BM42W36TR","download_json":"https://pith.science/pith/4NLTDTOLZR2EGHXQ5BM42W36TR.json","view_paper":"https://pith.science/paper/4NLTDTOL","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1601.00579&json=true","fetch_graph":"https://pith.science/api/pith-number/4NLTDTOLZR2EGHXQ5BM42W36TR/graph.json","fetch_events":"https://pith.science/api/pith-number/4NLTDTOLZR2EGHXQ5BM42W36TR/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/4NLTDTOLZR2EGHXQ5BM42W36TR/action/timestamp_anchor","attest_storage":"https://pith.science/pith/4NLTDTOLZR2EGHXQ5BM42W36TR/action/storage_attestation","attest_author":"https://pith.science/pith/4NLTDTOLZR2EGHXQ5BM42W36TR/action/author_attestation","sign_citation":"https://pith.science/pith/4NLTDTOLZR2EGHXQ5BM42W36TR/action/citation_signature","submit_replication":"https://pith.science/pith/4NLTDTOLZR2EGHXQ5BM42W36TR/action/replication_record"}},"created_at":"2026-05-18T00:31:29.087040+00:00","updated_at":"2026-05-18T00:31:29.087040+00:00"}