{"paper":{"title":"On the strong separation conjecture","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"D. Schaub (LAREMA, F Lucas (LAREMA, M. Spivakovsky (IMT, UA), UNAM)","submitted_at":"2018-02-26T15:19:40Z","abstract_excerpt":"This paper contains a partial result on the Pierce--Birkhoff conjecture on piece-wise polynomial functions defined by a finite collection {f 1,. .., f r} of polynomials. In the nineteen eighties, generalizing the problem from the polynomial ring to an artibtrary ring $\\Sigma$, J. Madden proved that the Pierce--Birkhoff conjecture for $\\Sigma$ is equivalent to a statement about an arbitrary pair of points $\\alpha$, $\\beta$ $\\in$ Sper $\\Sigma$ and their separating ideal < $\\alpha$, $\\beta$ >, we refer to this statement as the local Pierce-Birkhoff conjecture at $\\alpha$, $\\beta$. In [8] we intro"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1802.09389","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"}