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We are mainly interested in the case of pure cubic extensions, i.e. $a=0$ and $b\\in k\\setminus k^{3}$. We prove that if $\\op{deg}f=4$ and the variety $\\cal{S}_{f}$ contains a $k$-rational point $(x_{0},y_{0},z_{0},t_{0})$ with $f(t_{0})\\neq 0$, then $\\cal{"},"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":"1305.6242","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2013-05-27T14:44:30Z","cross_cats_sorted":[],"title_canon_sha256":"25783b02e18fda89fa6a2789ca403b6d6b5e8298eb54495af75967110d5160bc","abstract_canon_sha256":"ed6de960df56c1b454c34cbe92514df9f49bd640b66e8c5e66963a58214a6b0c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:24:51.778259Z","signature_b64":"DPuf4jgRAqoSyjX0rOgBYMXtS6WVagZDlBnWwPgzGzpi2oA66VlT3xqc9N1iKcK06mX3Eu2EwVpVq67jG2rfDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"315172df00e1f61ad6d1f4a6de5b3789bef30c0d639c9d17268448cb164eab11","last_reissued_at":"2026-05-18T03:24:51.777748Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:24:51.777748Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Rational solutions of certain Diophantine equations involving norms","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Maciej Ulas","submitted_at":"2013-05-27T14:44:30Z","abstract_excerpt":"In this note we present some results concerning the unirationality of the algebraic variety $\\cal{S}_{f}$ given by the equation \\begin{equation*} N_{K/k}(X_{1}+\\alpha X_{2}+\\alpha^2 X_{3})=f(t), \\end{equation*} where $k$ is a number field, $K=k(\\alpha)$, $\\alpha$ is a root of an irreducible polynomial $h(x)=x^3+ax+b\\in k[x]$ and $f\\in k[t]$. We are mainly interested in the case of pure cubic extensions, i.e. $a=0$ and $b\\in k\\setminus k^{3}$. We prove that if $\\op{deg}f=4$ and the variety $\\cal{S}_{f}$ contains a $k$-rational point $(x_{0},y_{0},z_{0},t_{0})$ with $f(t_{0})\\neq 0$, then $\\cal{"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1305.6242","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":"1305.6242","created_at":"2026-05-18T03:24:51.777825+00:00"},{"alias_kind":"arxiv_version","alias_value":"1305.6242v1","created_at":"2026-05-18T03:24:51.777825+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1305.6242","created_at":"2026-05-18T03:24:51.777825+00:00"},{"alias_kind":"pith_short_12","alias_value":"GFIXFXYA4H3B","created_at":"2026-05-18T12:27:45.050594+00:00"},{"alias_kind":"pith_short_16","alias_value":"GFIXFXYA4H3BVVWR","created_at":"2026-05-18T12:27:45.050594+00:00"},{"alias_kind":"pith_short_8","alias_value":"GFIXFXYA","created_at":"2026-05-18T12:27:45.050594+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/GFIXFXYA4H3BVVWR6STN4WZXRG","json":"https://pith.science/pith/GFIXFXYA4H3BVVWR6STN4WZXRG.json","graph_json":"https://pith.science/api/pith-number/GFIXFXYA4H3BVVWR6STN4WZXRG/graph.json","events_json":"https://pith.science/api/pith-number/GFIXFXYA4H3BVVWR6STN4WZXRG/events.json","paper":"https://pith.science/paper/GFIXFXYA"},"agent_actions":{"view_html":"https://pith.science/pith/GFIXFXYA4H3BVVWR6STN4WZXRG","download_json":"https://pith.science/pith/GFIXFXYA4H3BVVWR6STN4WZXRG.json","view_paper":"https://pith.science/paper/GFIXFXYA","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1305.6242&json=true","fetch_graph":"https://pith.science/api/pith-number/GFIXFXYA4H3BVVWR6STN4WZXRG/graph.json","fetch_events":"https://pith.science/api/pith-number/GFIXFXYA4H3BVVWR6STN4WZXRG/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/GFIXFXYA4H3BVVWR6STN4WZXRG/action/timestamp_anchor","attest_storage":"https://pith.science/pith/GFIXFXYA4H3BVVWR6STN4WZXRG/action/storage_attestation","attest_author":"https://pith.science/pith/GFIXFXYA4H3BVVWR6STN4WZXRG/action/author_attestation","sign_citation":"https://pith.science/pith/GFIXFXYA4H3BVVWR6STN4WZXRG/action/citation_signature","submit_replication":"https://pith.science/pith/GFIXFXYA4H3BVVWR6STN4WZXRG/action/replication_record"}},"created_at":"2026-05-18T03:24:51.777825+00:00","updated_at":"2026-05-18T03:24:51.777825+00:00"}