{"paper":{"title":"Rings of differentiable semialgebraic functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"E. Baro, J.M. Gamboa, Jos\\'e F. Fernando","submitted_at":"2019-08-20T10:07:48Z","abstract_excerpt":"In this work we analyze the main properties of the Zariski and maximal spectra of the ring ${\\mathcal S}^r(M)$ of differentiable semialgebraic functions of class ${\\mathcal C}^r$ on a semialgebraic set $M\\subset\\mathbb{R}^m$. Denote ${\\mathcal S}^0(M)$ the ring of semialgebraic functions on $M$ that admit a continuous extension to an open semialgebraic neighborhood of $M$ in $\\text{cl}(M)$. This ring is the real closure of ${\\mathcal S}^r(M)$. If $M$ is locally compact, the ring ${\\mathcal S}^r(M)$ enjoys a Lojasiewicz's Nullstellensatz, which becomes a crucial tool. Despite ${\\mathcal S}^r(M)"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1908.07257","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/1908.07257/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"}