{"paper":{"title":"Uniform definability of henselian valuation rings in the Macintyre language","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.LO"],"primary_cat":"math.AC","authors_text":"Alexander Prestel, Arno Fehm","submitted_at":"2014-08-20T20:41:25Z","abstract_excerpt":"We discuss definability of henselian valuation rings in the Macintyre language $\\mathcal{L}_{\\rm Mac}$, the language of rings expanded by n-th power predicates. In particular, we show that henselian valuation rings with finite or Hilbertian residue field are uniformly $\\exists$-$\\emptyset$-definable in $\\mathcal{L}_{\\rm Mac}$, and henselian valuation rings with value group $\\mathbb{Z}$ are uniformly $\\exists\\forall$-$\\emptyset$-definable in the ring language, but not uniformly $\\exists$-$\\emptyset$-definable in $\\mathcal{L}_{\\rm Mac}$. We apply these results to local fields $\\mathbb{Q}_p$ and "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1408.4816","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"}