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These cases include the determinant polynomial on the space of $n\\times n$ matrices, the Pfaffian on the space of skew-symmetric $2n\\times 2n$ matrices, and the determinant polynomial on the space of symmetric $n\\times n$ matrices. In particular, let $(V,F)$ be any pair among the following: $(\\textrm{Mat}_n, \\det)$, $(\\textrm{Skew}_{2n},\\textrm{Pff})$, and $(\\textrm{Sym}_n, \\det).$ We then obtain an asymptotic for $$\\pi_{V,F}(T)= \\#\\{v\\in V: \\max(|v_i|)\\leq T, F(v) \\text{ is prime}\\},"},"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":"2312.11445","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2023-12-18T18:49:50Z","cross_cats_sorted":["math.DS"],"title_canon_sha256":"f6a28d1c392449eb78f2c9802d44a73d1e7d9c47663579202cc9284fff9f6f48","abstract_canon_sha256":"cd7a00b93242193a7266b56b1f96a4f8711e595c962bf2f44a2aef92c35a60b2"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T07:25:34.435327Z","signature_b64":"VBdwC66PnLahwQXxxTKdkmOzTZ4lGsKCpqXPr3BDuakuadWf+GRudRCdYmJ0XlOXp1JyUsI8dOGCFRbp068OBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d40403ca0fbf493091391eb278ef6b43c763ca27ae4b1105ee190c32dafcb086","last_reissued_at":"2026-07-05T07:25:34.434834Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T07:25:34.434834Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Prime Number Theorems for Polynomials from Homogeneous Dynamics","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.DS"],"primary_cat":"math.NT","authors_text":"Giorgos Kotsovolis, Katharine Woo","submitted_at":"2023-12-18T18:49:50Z","abstract_excerpt":"We establish a new class of examples of the multivariate Bateman-Horn conjecture by using tools from dynamics. These cases include the determinant polynomial on the space of $n\\times n$ matrices, the Pfaffian on the space of skew-symmetric $2n\\times 2n$ matrices, and the determinant polynomial on the space of symmetric $n\\times n$ matrices. In particular, let $(V,F)$ be any pair among the following: $(\\textrm{Mat}_n, \\det)$, $(\\textrm{Skew}_{2n},\\textrm{Pff})$, and $(\\textrm{Sym}_n, \\det).$ We then obtain an asymptotic for $$\\pi_{V,F}(T)= \\#\\{v\\in V: \\max(|v_i|)\\leq T, F(v) \\text{ is prime}\\},"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2312.11445","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2312.11445/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2312.11445","created_at":"2026-07-05T07:25:34.434896+00:00"},{"alias_kind":"arxiv_version","alias_value":"2312.11445v1","created_at":"2026-07-05T07:25:34.434896+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2312.11445","created_at":"2026-07-05T07:25:34.434896+00:00"},{"alias_kind":"pith_short_12","alias_value":"2QCAHSQPX5ET","created_at":"2026-07-05T07:25:34.434896+00:00"},{"alias_kind":"pith_short_16","alias_value":"2QCAHSQPX5ETBEJZ","created_at":"2026-07-05T07:25:34.434896+00:00"},{"alias_kind":"pith_short_8","alias_value":"2QCAHSQP","created_at":"2026-07-05T07:25:34.434896+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":0,"sample":[{"citing_arxiv_id":"2604.25969","citing_title":"Effective Bounds for Singular Series in the Multivariate Bateman Horn Conjecture","ref_index":1,"is_internal_anchor":false}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/2QCAHSQPX5ETBEJZD2ZHR33LIP","json":"https://pith.science/pith/2QCAHSQPX5ETBEJZD2ZHR33LIP.json","graph_json":"https://pith.science/api/pith-number/2QCAHSQPX5ETBEJZD2ZHR33LIP/graph.json","events_json":"https://pith.science/api/pith-number/2QCAHSQPX5ETBEJZD2ZHR33LIP/events.json","paper":"https://pith.science/paper/2QCAHSQP"},"agent_actions":{"view_html":"https://pith.science/pith/2QCAHSQPX5ETBEJZD2ZHR33LIP","download_json":"https://pith.science/pith/2QCAHSQPX5ETBEJZD2ZHR33LIP.json","view_paper":"https://pith.science/paper/2QCAHSQP","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2312.11445&json=true","fetch_graph":"https://pith.science/api/pith-number/2QCAHSQPX5ETBEJZD2ZHR33LIP/graph.json","fetch_events":"https://pith.science/api/pith-number/2QCAHSQPX5ETBEJZD2ZHR33LIP/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/2QCAHSQPX5ETBEJZD2ZHR33LIP/action/timestamp_anchor","attest_storage":"https://pith.science/pith/2QCAHSQPX5ETBEJZD2ZHR33LIP/action/storage_attestation","attest_author":"https://pith.science/pith/2QCAHSQPX5ETBEJZD2ZHR33LIP/action/author_attestation","sign_citation":"https://pith.science/pith/2QCAHSQPX5ETBEJZD2ZHR33LIP/action/citation_signature","submit_replication":"https://pith.science/pith/2QCAHSQPX5ETBEJZD2ZHR33LIP/action/replication_record"}},"created_at":"2026-07-05T07:25:34.434896+00:00","updated_at":"2026-07-05T07:25:34.434896+00:00"}