{"paper":{"title":"On the Krull dimension of rings of semialgebraic functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"J.M. Gamboa, Jos\\'e F. Fernando","submitted_at":"2013-06-18T09:05:53Z","abstract_excerpt":"Let $R$ be a real closed field and let ${\\mathcal S}(M)$ be the ring of (continuous) semialgebraic functions on a semialgebraic set $M\\subset R^n$ and let ${\\mathcal S}^*(M)$ be its subring of bounded semialgebraic functions. In this work we introduce the concept of \\em semialgebraic depth \\em of a prime ideal $\\gtp$ of ${\\mathcal S}(M)$ in order to provide an elementary proof of the finiteness of the Krull dimension of the rings ${\\mathcal S}(M)$ and ${\\mathcal S}^*(M)$, inspired in the classical way of doing to compute the dimension of a ring of polynomials on a complex algebraic set and wit"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1306.4109","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"}