{"paper":{"title":"First-order definitions of rings of integral functions over algebraic extensions of function fields and undecidability","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.LO"],"primary_cat":"math.NT","authors_text":"Alexandra Shlapentokh, Caleb Springer","submitted_at":"2024-11-22T14:18:10Z","abstract_excerpt":"In this paper, we study questions of definability and decidability for infinite algebraic extensions ${\\bf K}$ of $\\mathbb{F}_p(t)$ and their subrings of $\\mathcal{S}$-integral functions. We focus on fields ${\\bf K}$ satisfying a local property which we call $q$-boundedness. This can be considered a function field analogue of prior work of the first author which considered algebraic extensions of $\\mathbb{Q}$. One simple consequence of our work states that if ${\\bf K}$ is a $q$-bounded Galois extension of $\\mathbb{F}_p(t)$, then for infinitely many non-constant $u$ the integral closure $\\mathc"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2411.14960","kind":"arxiv","version":2},"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/2411.14960/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"}