{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2011:NKNTIQZA7FGUIQAFGXEPPRL3L6","short_pith_number":"pith:NKNTIQZA","schema_version":"1.0","canonical_sha256":"6a9b344320f94d44400535c8f7c57b5fb0d1c374ce8dbbfe2057dd997d923dc0","source":{"kind":"arxiv","id":"1106.0792","version":1},"attestation_state":"computed","paper":{"title":"Polynomial maps with invertible sums of Jacobian matrices and of directional Derivatives","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Hongbo Guo, Michiel de Bondt, Xiankun Du, Xiaosong Sun","submitted_at":"2011-06-04T06:46:45Z","abstract_excerpt":"Let $F: C^n \\rightarrow C^m$ be a polynomial map with $degF=d \\geq 2$. We prove that $F$ is invertible if $m = n$ and $\\sum^{d-1}_{i=1} JF(\\alpha_i)$ is invertible for all $i$, which is trivially the case for invertible quadratic maps. More generally, we prove that for affine lines $L = \\{\\beta + \\mu \\gamma | \\mu \\in C\\} \\subseteq C^n$ ($\\gamma \\ne 0$), $F|_L$ is linearly rectifiable, if and only if $\\sum^{d-1}_{i=1} JF(\\alpha_i) \\cdot \\gamma \\ne 0$ for all $\\alpha_i \\in L$. This appears to be the case for all affine lines $L$ when $F$ is injective and $d \\le 3$. We also prove that if $m = n$ "},"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":"1106.0792","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2011-06-04T06:46:45Z","cross_cats_sorted":[],"title_canon_sha256":"62a241984680ad527a49bfcd75299f8a202046bfa31e947caef89c03e61aee28","abstract_canon_sha256":"aa49fd8462a14300bb3131e519a3744514a0bef99a9263f8bc24f2c5f9d0e852"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:09:23.229658Z","signature_b64":"K52RmXB3T0ZTiLL7+UL/74USmGwyVexOBbSs9YHFAb6xonmbKVlWcDvMsWF4jE6G10D+GHeomUAPhiJXBPpbDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6a9b344320f94d44400535c8f7c57b5fb0d1c374ce8dbbfe2057dd997d923dc0","last_reissued_at":"2026-05-18T03:09:23.228916Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:09:23.228916Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Polynomial maps with invertible sums of Jacobian matrices and of directional Derivatives","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Hongbo Guo, Michiel de Bondt, Xiankun Du, Xiaosong Sun","submitted_at":"2011-06-04T06:46:45Z","abstract_excerpt":"Let $F: C^n \\rightarrow C^m$ be a polynomial map with $degF=d \\geq 2$. We prove that $F$ is invertible if $m = n$ and $\\sum^{d-1}_{i=1} JF(\\alpha_i)$ is invertible for all $i$, which is trivially the case for invertible quadratic maps. More generally, we prove that for affine lines $L = \\{\\beta + \\mu \\gamma | \\mu \\in C\\} \\subseteq C^n$ ($\\gamma \\ne 0$), $F|_L$ is linearly rectifiable, if and only if $\\sum^{d-1}_{i=1} JF(\\alpha_i) \\cdot \\gamma \\ne 0$ for all $\\alpha_i \\in L$. This appears to be the case for all affine lines $L$ when $F$ is injective and $d \\le 3$. We also prove that if $m = n$ "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1106.0792","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1106.0792","created_at":"2026-05-18T03:09:23.229041+00:00"},{"alias_kind":"arxiv_version","alias_value":"1106.0792v1","created_at":"2026-05-18T03:09:23.229041+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1106.0792","created_at":"2026-05-18T03:09:23.229041+00:00"},{"alias_kind":"pith_short_12","alias_value":"NKNTIQZA7FGU","created_at":"2026-05-18T12:26:37.096874+00:00"},{"alias_kind":"pith_short_16","alias_value":"NKNTIQZA7FGUIQAF","created_at":"2026-05-18T12:26:37.096874+00:00"},{"alias_kind":"pith_short_8","alias_value":"NKNTIQZA","created_at":"2026-05-18T12:26:37.096874+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/NKNTIQZA7FGUIQAFGXEPPRL3L6","json":"https://pith.science/pith/NKNTIQZA7FGUIQAFGXEPPRL3L6.json","graph_json":"https://pith.science/api/pith-number/NKNTIQZA7FGUIQAFGXEPPRL3L6/graph.json","events_json":"https://pith.science/api/pith-number/NKNTIQZA7FGUIQAFGXEPPRL3L6/events.json","paper":"https://pith.science/paper/NKNTIQZA"},"agent_actions":{"view_html":"https://pith.science/pith/NKNTIQZA7FGUIQAFGXEPPRL3L6","download_json":"https://pith.science/pith/NKNTIQZA7FGUIQAFGXEPPRL3L6.json","view_paper":"https://pith.science/paper/NKNTIQZA","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1106.0792&json=true","fetch_graph":"https://pith.science/api/pith-number/NKNTIQZA7FGUIQAFGXEPPRL3L6/graph.json","fetch_events":"https://pith.science/api/pith-number/NKNTIQZA7FGUIQAFGXEPPRL3L6/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/NKNTIQZA7FGUIQAFGXEPPRL3L6/action/timestamp_anchor","attest_storage":"https://pith.science/pith/NKNTIQZA7FGUIQAFGXEPPRL3L6/action/storage_attestation","attest_author":"https://pith.science/pith/NKNTIQZA7FGUIQAFGXEPPRL3L6/action/author_attestation","sign_citation":"https://pith.science/pith/NKNTIQZA7FGUIQAFGXEPPRL3L6/action/citation_signature","submit_replication":"https://pith.science/pith/NKNTIQZA7FGUIQAFGXEPPRL3L6/action/replication_record"}},"created_at":"2026-05-18T03:09:23.229041+00:00","updated_at":"2026-05-18T03:09:23.229041+00:00"}