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Umberto Zannier showed that there exists a matrix in $\\operatorname{SL}_3(K[x,y,z])$ with first row $(x,y,z)$ for $K=\\mathbb Q_p$, the field of $p$-adic numbers for an odd prime $p$, or more generally, if $-1$ is a sum of two squares in $K$. The case $K=\\mathbb Q_2$ remained open and was subsequently posed and discussed by Zannier with numerous researchers, thereby bringing the problem to broader attention.\n  In 2025, Alexey Ananyevskiy and Marc Levine showed that such a matrix exists ","authors_text":"Peter M\\\"uller","cross_cats":["math.AG"],"headline":"Explicit formulas produce a matrix in SL_3 over the unit sphere ring from four squares summing to minus one.","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2026-05-14T22:29:02Z","title":"Explicitly combing hedgehogs over fields of Stufe 4"},"references":{"count":4,"internal_anchors":0,"resolved_work":4,"sample":[{"cited_arxiv_id":"","doi":"10.2140/ant","is_internal_anchor":false,"ref_index":1,"title":"Combing a hedgehog over a field","work_id":"e917cd6b-0e20-44c9-8529-8b698464506d","year":2025},{"cited_arxiv_id":"","doi":"10.1145/3452143.3465545","is_internal_anchor":false,"ref_index":2,"title":"msolve: A Library for Solv- ing Polynomial Systems","work_id":"d193af2a-840e-426f-a875-295b7bd6314f","year":2021},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":3,"title":"Müller.SageMath verification script.https://ypfmde.github.io/ verify_combing.html","work_id":"dedd0c48-c1eb-45d2-a6d5-eda9b0da3163","year":2026},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":4,"title":"https://www.sagemath.org","work_id":"f4147711-4ec7-4c67-a9e4-1a37cee43f7b","year":2022}],"snapshot_sha256":"34d0a737257eed923b83bf4e50323d312a77f328e1b51b504c2ddc811f142f04"},"source":{"id":"2605.15452","kind":"arxiv","version":1},"verdict":{"created_at":"2026-05-19T14:24:23.929754Z","id":"3eeaf89d-52d2-4b53-953c-d3279eb6e072","model_set":{"reader":"grok-4.3"},"one_line_summary":"Explicit matrix in SL_3(K[x,y,z]) with first row (x,y,z) for any field K of Stufe at most 4, expressed in terms of a,b,c,d satisfying a²+b²+c²+d²=-1.","pipeline_version":"pith-pipeline@v0.9.0","pith_extraction_headline":"Explicit formulas produce a matrix in SL_3 over the unit sphere ring from four squares summing to minus one.","strongest_claim":"We construct an explicit example in terms of a,b,c,d of a matrix in SL_3(K[x,y,z]) with first row (x,y,z) whenever a² + b² + c² + d² = -1 in K.","weakest_assumption":"The algebraic identities that define the matrix entries from a,b,c,d remain valid inside the quotient ring K[x,y,z]/(x²+y²+z²-1) and produce determinant 1; this is invoked when the explicit formulas are substituted and simplified."}},"verdict_id":"3eeaf89d-52d2-4b53-953c-d3279eb6e072"}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:71e03fe6bb4bc570bbe7a3e15911a2bb8a1d702e688b4a58fc5e8fd0ccd81d22","target":"record","created_at":"2026-05-20T00:00:59Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"f7c504951f1de88b8024a0e201ce3d5e0396317ab354865831294a83aa233a15","cross_cats_sorted":["math.AG"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2026-05-14T22:29:02Z","title_canon_sha256":"2b452565bd7fd92a06a11bb264ad7fa58892fc738d408e3d31bdfc1731dfe24f"},"schema_version":"1.0","source":{"id":"2605.15452","kind":"arxiv","version":1}},"canonical_sha256":"9a2f6cac1a103e3e9d929b0409ab1b6390eab98b7fd2a277e555d0606473fa63","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"9a2f6cac1a103e3e9d929b0409ab1b6390eab98b7fd2a277e555d0606473fa63","first_computed_at":"2026-05-20T00:00:59.347040Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-20T00:00:59.347040Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"YCfqjYFSOnkkPKBOr3mabdsJC1mgvt0oWy7zZmv14moUrE9G9ub/Jw1iWCDo+BAdE8snRDdtnnf0aELYp37MBg==","signature_status":"signed_v1","signed_at":"2026-05-20T00:00:59.347652Z","signed_message":"canonical_sha256_bytes"},"source_id":"2605.15452","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:71e03fe6bb4bc570bbe7a3e15911a2bb8a1d702e688b4a58fc5e8fd0ccd81d22","sha256:4c2ee61d13a525c05e53a8221c79ec9c691f5a3186e4a811b2d7e1d2b688ce51"],"state_sha256":"60fd4065e22bb2ab5d1206e971cfbbc49ce52f2d59674c8615d34b94ad356d2f"}