{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2013:JXFXEUMQ6T55BPCFNFV6YZOQF4","short_pith_number":"pith:JXFXEUMQ","schema_version":"1.0","canonical_sha256":"4dcb725190f4fbd0bc45696bec65d02f0a4ac221d5cd73d5a5c53550e671e712","source":{"kind":"arxiv","id":"1302.2358","version":2},"attestation_state":"computed","paper":{"title":"A Real Nullstellensatz for Free Modules","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Jaka Cimpric","submitted_at":"2013-02-10T20:23:27Z","abstract_excerpt":"Let $A$ be the algebra of all $n \\times n$ matrices with entries from $\\RR[x_1,\\ldots,x_d]$ and let $G_1,\\ldots,G_m,F \\in A$. We will show that $F(a)v=0$ for every $a \\in \\RR^d$ and $v \\in \\RR^n$ such that $G_i(a)v=0$ for all $i$ if and only if $F$ belongs to the smallest real left ideal of $A$ which contains $G_1,\\ldots,G_m$. Here a left ideal $J$ of $A$ is real if for every $H_1,\\ldots,H_k \\in A$ such that $H_1^T H_1+\\ldots+H_k^T H_k \\in J+J^T$ we have that $H_1,\\ldots,H_k \\in J$. We call this result the one-sided Real Nullstellensatz for matrix polynomials. We first prove by induction on $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":"1302.2358","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2013-02-10T20:23:27Z","cross_cats_sorted":[],"title_canon_sha256":"92b338935042a1d1096e9cc87116a50e86c0e8e53d96b3989e58c8c281cf24e4","abstract_canon_sha256":"2b21c0e90bded866a06fa39ff0f8aafca52ca24ff10871afb799b553779980b2"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:17:55.712131Z","signature_b64":"S22qJNZRXKHzIGU8XRfat7UqvEsEpx514WdHvaFmvqeYiVySQZGGihQHnpcIF2ehh4/+/cpiM/5qgoGiHACJCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4dcb725190f4fbd0bc45696bec65d02f0a4ac221d5cd73d5a5c53550e671e712","last_reissued_at":"2026-05-18T00:17:55.711604Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:17:55.711604Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A Real Nullstellensatz for Free Modules","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Jaka Cimpric","submitted_at":"2013-02-10T20:23:27Z","abstract_excerpt":"Let $A$ be the algebra of all $n \\times n$ matrices with entries from $\\RR[x_1,\\ldots,x_d]$ and let $G_1,\\ldots,G_m,F \\in A$. We will show that $F(a)v=0$ for every $a \\in \\RR^d$ and $v \\in \\RR^n$ such that $G_i(a)v=0$ for all $i$ if and only if $F$ belongs to the smallest real left ideal of $A$ which contains $G_1,\\ldots,G_m$. Here a left ideal $J$ of $A$ is real if for every $H_1,\\ldots,H_k \\in A$ such that $H_1^T H_1+\\ldots+H_k^T H_k \\in J+J^T$ we have that $H_1,\\ldots,H_k \\in J$. We call this result the one-sided Real Nullstellensatz for matrix polynomials. We first prove by induction on $n"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1302.2358","kind":"arxiv","version":2},"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":"1302.2358","created_at":"2026-05-18T00:17:55.711687+00:00"},{"alias_kind":"arxiv_version","alias_value":"1302.2358v2","created_at":"2026-05-18T00:17:55.711687+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1302.2358","created_at":"2026-05-18T00:17:55.711687+00:00"},{"alias_kind":"pith_short_12","alias_value":"JXFXEUMQ6T55","created_at":"2026-05-18T12:27:49.015174+00:00"},{"alias_kind":"pith_short_16","alias_value":"JXFXEUMQ6T55BPCF","created_at":"2026-05-18T12:27:49.015174+00:00"},{"alias_kind":"pith_short_8","alias_value":"JXFXEUMQ","created_at":"2026-05-18T12:27:49.015174+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/JXFXEUMQ6T55BPCFNFV6YZOQF4","json":"https://pith.science/pith/JXFXEUMQ6T55BPCFNFV6YZOQF4.json","graph_json":"https://pith.science/api/pith-number/JXFXEUMQ6T55BPCFNFV6YZOQF4/graph.json","events_json":"https://pith.science/api/pith-number/JXFXEUMQ6T55BPCFNFV6YZOQF4/events.json","paper":"https://pith.science/paper/JXFXEUMQ"},"agent_actions":{"view_html":"https://pith.science/pith/JXFXEUMQ6T55BPCFNFV6YZOQF4","download_json":"https://pith.science/pith/JXFXEUMQ6T55BPCFNFV6YZOQF4.json","view_paper":"https://pith.science/paper/JXFXEUMQ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1302.2358&json=true","fetch_graph":"https://pith.science/api/pith-number/JXFXEUMQ6T55BPCFNFV6YZOQF4/graph.json","fetch_events":"https://pith.science/api/pith-number/JXFXEUMQ6T55BPCFNFV6YZOQF4/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/JXFXEUMQ6T55BPCFNFV6YZOQF4/action/timestamp_anchor","attest_storage":"https://pith.science/pith/JXFXEUMQ6T55BPCFNFV6YZOQF4/action/storage_attestation","attest_author":"https://pith.science/pith/JXFXEUMQ6T55BPCFNFV6YZOQF4/action/author_attestation","sign_citation":"https://pith.science/pith/JXFXEUMQ6T55BPCFNFV6YZOQF4/action/citation_signature","submit_replication":"https://pith.science/pith/JXFXEUMQ6T55BPCFNFV6YZOQF4/action/replication_record"}},"created_at":"2026-05-18T00:17:55.711687+00:00","updated_at":"2026-05-18T00:17:55.711687+00:00"}