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The proof depends on showing that if $K$ is infinite and $n$ is a positive integer, there exists a positive integer C(n), independent of $N$, such that any $n$ forms of degree at most 2 in $R$ are contained in a subring of $R$ generated over $K$ by at most $t \\leq C(n)$ forms $G_1, \\,..., \\, G_t$ of degree 1 or 2 such that $G_1, \\,..."},"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":"1106.0839","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2011-06-04T16:49:16Z","cross_cats_sorted":[],"title_canon_sha256":"408f902fcfdbb2ffa39c2bbe5afc1905fc3fa727c062803b943bc06ad589b334","abstract_canon_sha256":"e20fb08f22459ade248ce78385be320ebcb5a72754fd10b9833b9882d89d7d9e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:20:37.346104Z","signature_b64":"Zd1J2NbWoCbYJpgRB6MgxcSxIUcqsJFbXFDrDQz36bRAF72FVvnG6EcWDiOh8hmh+3bVSj2pc72gPaEcIfkSCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8bd3f599c609c6cbe69a9803275d5de6429e87185a9bf3ff9c5615ca5f84eedc","last_reissued_at":"2026-05-18T04:20:37.345144Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:20:37.345144Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Ideals Generated by Quadratic Polynomials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Melvin Hochster, Tigran Ananyan","submitted_at":"2011-06-04T16:49:16Z","abstract_excerpt":"Let $R$ be a polynomial ring in $N$ variables over an arbitrary field $K$ and let $I$ be an ideal of $R$ generated by $n$ polynomials of degree at most 2. 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The proof depends on showing that if $K$ is infinite and $n$ is a positive integer, there exists a positive integer C(n), independent of $N$, such that any $n$ forms of degree at most 2 in $R$ are contained in a subring of $R$ generated over $K$ by at most $t \\leq C(n)$ forms $G_1, \\,..., \\, G_t$ of degree 1 or 2 such that $G_1, \\,..."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1106.0839","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":"1106.0839","created_at":"2026-05-18T04:20:37.345262+00:00"},{"alias_kind":"arxiv_version","alias_value":"1106.0839v1","created_at":"2026-05-18T04:20:37.345262+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1106.0839","created_at":"2026-05-18T04:20:37.345262+00:00"},{"alias_kind":"pith_short_12","alias_value":"RPJ7LGOGBHDM","created_at":"2026-05-18T12:26:41.206345+00:00"},{"alias_kind":"pith_short_16","alias_value":"RPJ7LGOGBHDMXZU2","created_at":"2026-05-18T12:26:41.206345+00:00"},{"alias_kind":"pith_short_8","alias_value":"RPJ7LGOG","created_at":"2026-05-18T12:26:41.206345+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/RPJ7LGOGBHDMXZU2TABSOXK54Z","json":"https://pith.science/pith/RPJ7LGOGBHDMXZU2TABSOXK54Z.json","graph_json":"https://pith.science/api/pith-number/RPJ7LGOGBHDMXZU2TABSOXK54Z/graph.json","events_json":"https://pith.science/api/pith-number/RPJ7LGOGBHDMXZU2TABSOXK54Z/events.json","paper":"https://pith.science/paper/RPJ7LGOG"},"agent_actions":{"view_html":"https://pith.science/pith/RPJ7LGOGBHDMXZU2TABSOXK54Z","download_json":"https://pith.science/pith/RPJ7LGOGBHDMXZU2TABSOXK54Z.json","view_paper":"https://pith.science/paper/RPJ7LGOG","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1106.0839&json=true","fetch_graph":"https://pith.science/api/pith-number/RPJ7LGOGBHDMXZU2TABSOXK54Z/graph.json","fetch_events":"https://pith.science/api/pith-number/RPJ7LGOGBHDMXZU2TABSOXK54Z/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/RPJ7LGOGBHDMXZU2TABSOXK54Z/action/timestamp_anchor","attest_storage":"https://pith.science/pith/RPJ7LGOGBHDMXZU2TABSOXK54Z/action/storage_attestation","attest_author":"https://pith.science/pith/RPJ7LGOGBHDMXZU2TABSOXK54Z/action/author_attestation","sign_citation":"https://pith.science/pith/RPJ7LGOGBHDMXZU2TABSOXK54Z/action/citation_signature","submit_replication":"https://pith.science/pith/RPJ7LGOGBHDMXZU2TABSOXK54Z/action/replication_record"}},"created_at":"2026-05-18T04:20:37.345262+00:00","updated_at":"2026-05-18T04:20:37.345262+00:00"}