{"paper":{"title":"On the substitution theorem for rings of semialgebraic functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Jose F. Fernando","submitted_at":"2013-09-15T10:21:02Z","abstract_excerpt":"Let $R\\subset F$ be an extension of real closed fields and ${\\mathcal S}(M,R)$ the ring of (continuous) semialgebraic functions on a semialgebraic set $M\\subset R^n$. We prove that every $R$-homomorphism $\\varphi:{\\mathcal S}(M,R)\\to F$ is essentially the evaluation homomorphism at a certain point $p\\in F^n$ \\em adjacent \\em to the extended semialgebraic set $M_F$. This type of result is commonly known in Real Algebra as Substitution Theorem. In case $M$ is locally closed, the results are neat while the non locally closed case requires a more subtle approach and some constructions (weak contin"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1309.3743","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"}