{"paper":{"title":"$\\mathbb Q\\setminus\\mathbb Z$ is diophantine over $\\mathbb Q$ with $7$ unknowns","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.LO"],"primary_cat":"math.NT","authors_text":"Zhi-Wei Sun","submitted_at":"2026-07-30T17:56:28Z","abstract_excerpt":"In 2016 J. Koenigsmann proved that $\\mathbb Q\\setminus\\mathbb Z$ is diophantine over $\\mathbb Q$, i.e., there is a polynomial $P(t,x_1,\\ldots,x_{n})\\in\\mathbb Z[t,x_1,\\ldots,x_{n}]$ such that for any rational number $t$ we have $$t\\not\\in\\mathbb Z\\iff \\exists x_1,\\ldots,x_{n}\\in\\mathbb Q\\,[P(t,x_1,\\ldots,x_{n})=0].$$ In this paper we show that we may take $n=7$ which improves the previous record $n=10$ obtained by Daans in 2024. (Actually we even extend this to any global field.) This, together with a previous result of Z.-W. Sun, implies that there is no algorithm to decide for any $F(x_1,\\ld"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.28606","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/2607.28606/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"}