{"paper":{"title":"Le compl\\'ementaire des puissances $n$-i\\`emes dans un corps de nombres est un ensemble diophantien","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG"],"primary_cat":"math.NT","authors_text":"Jan Van Geel, Jean-Louis Colliot-Th\\'el\\`ene","submitted_at":"2014-01-05T17:31:53Z","abstract_excerpt":"Given a number field $k$ and a positive integer $n$, there exists an algebraic variety $X$ over $k$ and a function $f$ on $X$ whose set of values $f(X(k))$ on the set of $k$-points of $X$ is the complement in $k$ of the set of $n$-th powers. This result had been proved by B. Poonen (2009) for $n$ a power of $2$. For $n$ arbitrary, under Schinzel's hypothesis, it has been given a conditional proof by T. V\\'arilly-Alvarado and B. Viray (2012). Instead of Schinzel's hypothesis, we use \"Salberger's trick\", as developed in papers of Skorobogatov, Swinnerton-Dyer and one of the authors."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1401.0915","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"}