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For non-negative integers $m$ and $d$ we let $mW\\oplus d W^*$ denote the representation of $G$ given by the direct sum of $m$ vectors and $d$ covectors. We exhibit a minimal set of homogenous invariant polynomials $\\{\\ell_1,\\ell_{2},\\dots,\\ell_{(m+d)n}\\}\\subseteq \\mathbb{F}_q[mW\\oplus d W^*]^G$ such that $\\mathbb{F}_q(mW\\oplus d W^*)^G=\\mathbb{F}_q(\\el"},"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":"1708.01593","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2017-08-04T17:23:44Z","cross_cats_sorted":[],"title_canon_sha256":"a5c6e2f85f93876e7158d1bd81f59109bdab4ac54285f5dc26e4c7253e1f70ba","abstract_canon_sha256":"22bdcbfbbca26f654c7e277eb5ea944d49d5d7831ad7afb91f256aee22f92e90"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:35:46.150650Z","signature_b64":"kFqKtBa7gwrv7kGsGQkY7FvGADKLesV4vYWsK1n436uw2LmQrHMM6xof+SshPYk/RsUCoWhR0acOQ8PNtcE4Aw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"52e366a992b97b85bbc50dcd7cf7bfd5da787d01617611a4e4845c6b3396cf08","last_reissued_at":"2026-05-18T00:35:46.150096Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:35:46.150096Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On invariant fields of vectors and covectors","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"David L. Wehlau, Yin Chen","submitted_at":"2017-08-04T17:23:44Z","abstract_excerpt":"Let ${\\mathbb{F}_{q}}$ be the finite field of order $q$. Let $G$ be one of the three groups ${\\rm GL}(n, \\mathbb{F}_q)$, ${\\rm SL}(n, \\mathbb{F}_q)$ or ${\\rm U}(n, \\mathbb{F}_q)$ and let $W$ be the standard $n$-dimensional representation of $G$. For non-negative integers $m$ and $d$ we let $mW\\oplus d W^*$ denote the representation of $G$ given by the direct sum of $m$ vectors and $d$ covectors. We exhibit a minimal set of homogenous invariant polynomials $\\{\\ell_1,\\ell_{2},\\dots,\\ell_{(m+d)n}\\}\\subseteq \\mathbb{F}_q[mW\\oplus d W^*]^G$ such that $\\mathbb{F}_q(mW\\oplus d W^*)^G=\\mathbb{F}_q(\\el"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1708.01593","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":"1708.01593","created_at":"2026-05-18T00:35:46.150178+00:00"},{"alias_kind":"arxiv_version","alias_value":"1708.01593v2","created_at":"2026-05-18T00:35:46.150178+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1708.01593","created_at":"2026-05-18T00:35:46.150178+00:00"},{"alias_kind":"pith_short_12","alias_value":"KLRWNKMSXF5Y","created_at":"2026-05-18T12:31:24.725408+00:00"},{"alias_kind":"pith_short_16","alias_value":"KLRWNKMSXF5YLO6F","created_at":"2026-05-18T12:31:24.725408+00:00"},{"alias_kind":"pith_short_8","alias_value":"KLRWNKMS","created_at":"2026-05-18T12:31:24.725408+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/KLRWNKMSXF5YLO6FBXGXZ5572X","json":"https://pith.science/pith/KLRWNKMSXF5YLO6FBXGXZ5572X.json","graph_json":"https://pith.science/api/pith-number/KLRWNKMSXF5YLO6FBXGXZ5572X/graph.json","events_json":"https://pith.science/api/pith-number/KLRWNKMSXF5YLO6FBXGXZ5572X/events.json","paper":"https://pith.science/paper/KLRWNKMS"},"agent_actions":{"view_html":"https://pith.science/pith/KLRWNKMSXF5YLO6FBXGXZ5572X","download_json":"https://pith.science/pith/KLRWNKMSXF5YLO6FBXGXZ5572X.json","view_paper":"https://pith.science/paper/KLRWNKMS","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1708.01593&json=true","fetch_graph":"https://pith.science/api/pith-number/KLRWNKMSXF5YLO6FBXGXZ5572X/graph.json","fetch_events":"https://pith.science/api/pith-number/KLRWNKMSXF5YLO6FBXGXZ5572X/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/KLRWNKMSXF5YLO6FBXGXZ5572X/action/timestamp_anchor","attest_storage":"https://pith.science/pith/KLRWNKMSXF5YLO6FBXGXZ5572X/action/storage_attestation","attest_author":"https://pith.science/pith/KLRWNKMSXF5YLO6FBXGXZ5572X/action/author_attestation","sign_citation":"https://pith.science/pith/KLRWNKMSXF5YLO6FBXGXZ5572X/action/citation_signature","submit_replication":"https://pith.science/pith/KLRWNKMSXF5YLO6FBXGXZ5572X/action/replication_record"}},"created_at":"2026-05-18T00:35:46.150178+00:00","updated_at":"2026-05-18T00:35:46.150178+00:00"}