{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2015:NXM353DYSDFEXIZOV3ZGGLDIJ4","short_pith_number":"pith:NXM353DY","schema_version":"1.0","canonical_sha256":"6dd9beec7890ca4ba32eaef2632c684f2b8c52902a6d6f330ae14e5a2490d8c1","source":{"kind":"arxiv","id":"1512.03393","version":1},"attestation_state":"computed","paper":{"title":"Polynomial degree bounds for matrix semi-invariants","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CC"],"primary_cat":"math.RT","authors_text":"Harm Derksen, Visu Makam","submitted_at":"2015-12-10T20:12:47Z","abstract_excerpt":"We study the left-right action of $\\operatorname{SL}_n \\times \\operatorname{SL}_n$ on $m$-tuples of $n \\times n$ matrices with entries in an infinite field $K$. We show that invariants of degree $n^2- n$ define the null cone. Consequently, invariants of degree $\\leq n^6$ generate the ring of invariants if $\\operatorname{char}(K)=0$. We also prove that for $m \\gg 0$, invariants of degree at least $n\\lfloor \\sqrt{n+1}\\rfloor$ are required to define the null cone. We generalize our results to matrix invariants of $m$-tuples of $p\\times q$ matrices, and to rings of semi-invariants for quivers. For"},"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":"1512.03393","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2015-12-10T20:12:47Z","cross_cats_sorted":["cs.CC"],"title_canon_sha256":"4567678a5a66c9096d95c5a434c64e4a351b99fca3a3a426e17ac1dab7711a2d","abstract_canon_sha256":"bf89e0c9262b7dafeabe3cfbf7eab94c65ddd88e037769b22dc38b0740591002"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:24:37.305345Z","signature_b64":"IAFTywCj/praxe19oNgqNqUiwF2FhCT7A0fTFoe6kWxe+oANq8h+Mr9IKBHqB170GkHHNRL1DmmPkUhqJxUyAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6dd9beec7890ca4ba32eaef2632c684f2b8c52902a6d6f330ae14e5a2490d8c1","last_reissued_at":"2026-05-18T01:24:37.304927Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:24:37.304927Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Polynomial degree bounds for matrix semi-invariants","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CC"],"primary_cat":"math.RT","authors_text":"Harm Derksen, Visu Makam","submitted_at":"2015-12-10T20:12:47Z","abstract_excerpt":"We study the left-right action of $\\operatorname{SL}_n \\times \\operatorname{SL}_n$ on $m$-tuples of $n \\times n$ matrices with entries in an infinite field $K$. We show that invariants of degree $n^2- n$ define the null cone. Consequently, invariants of degree $\\leq n^6$ generate the ring of invariants if $\\operatorname{char}(K)=0$. We also prove that for $m \\gg 0$, invariants of degree at least $n\\lfloor \\sqrt{n+1}\\rfloor$ are required to define the null cone. We generalize our results to matrix invariants of $m$-tuples of $p\\times q$ matrices, and to rings of semi-invariants for quivers. For"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1512.03393","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":"1512.03393","created_at":"2026-05-18T01:24:37.304991+00:00"},{"alias_kind":"arxiv_version","alias_value":"1512.03393v1","created_at":"2026-05-18T01:24:37.304991+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1512.03393","created_at":"2026-05-18T01:24:37.304991+00:00"},{"alias_kind":"pith_short_12","alias_value":"NXM353DYSDFE","created_at":"2026-05-18T12:29:34.919912+00:00"},{"alias_kind":"pith_short_16","alias_value":"NXM353DYSDFEXIZO","created_at":"2026-05-18T12:29:34.919912+00:00"},{"alias_kind":"pith_short_8","alias_value":"NXM353DY","created_at":"2026-05-18T12:29:34.919912+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":0,"sample":[{"citing_arxiv_id":"2605.08085","citing_title":"Fermionic trace relations and supersymmetric indices at finite $N$","ref_index":75,"is_internal_anchor":false}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/NXM353DYSDFEXIZOV3ZGGLDIJ4","json":"https://pith.science/pith/NXM353DYSDFEXIZOV3ZGGLDIJ4.json","graph_json":"https://pith.science/api/pith-number/NXM353DYSDFEXIZOV3ZGGLDIJ4/graph.json","events_json":"https://pith.science/api/pith-number/NXM353DYSDFEXIZOV3ZGGLDIJ4/events.json","paper":"https://pith.science/paper/NXM353DY"},"agent_actions":{"view_html":"https://pith.science/pith/NXM353DYSDFEXIZOV3ZGGLDIJ4","download_json":"https://pith.science/pith/NXM353DYSDFEXIZOV3ZGGLDIJ4.json","view_paper":"https://pith.science/paper/NXM353DY","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1512.03393&json=true","fetch_graph":"https://pith.science/api/pith-number/NXM353DYSDFEXIZOV3ZGGLDIJ4/graph.json","fetch_events":"https://pith.science/api/pith-number/NXM353DYSDFEXIZOV3ZGGLDIJ4/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/NXM353DYSDFEXIZOV3ZGGLDIJ4/action/timestamp_anchor","attest_storage":"https://pith.science/pith/NXM353DYSDFEXIZOV3ZGGLDIJ4/action/storage_attestation","attest_author":"https://pith.science/pith/NXM353DYSDFEXIZOV3ZGGLDIJ4/action/author_attestation","sign_citation":"https://pith.science/pith/NXM353DYSDFEXIZOV3ZGGLDIJ4/action/citation_signature","submit_replication":"https://pith.science/pith/NXM353DYSDFEXIZOV3ZGGLDIJ4/action/replication_record"}},"created_at":"2026-05-18T01:24:37.304991+00:00","updated_at":"2026-05-18T01:24:37.304991+00:00"}