{"paper":{"title":"The Proper Basis for Polynomial Ideals","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":["cs.SC","math.AG"],"primary_cat":"math.AC","authors_text":"Sheng-Ming Ma","submitted_at":"2021-01-10T06:29:49Z","abstract_excerpt":"We define a new type of ideal basis called the proper basis that improves both Gr\\\"obner basis and Buchberger's algorithm. Let $x_1$ be the least variable of a monomial ordering in a polynomial ring $K[x_1,\\dotsc,x_n]$ over a field $K$. The Gr\\\"obner basis of a zero-dimensional polynomial ideal contains a univariate polynomial in $x_1$. The proper basis is defined and computed in the variables $\\tilde{\\bm{x}}:=(x_2,\\dotsc,x_n)$ with $x_1$ serving as a parameter in the algebra $K[x_1][\\tilde{\\bm{x}}]$. Its algorithm is more efficient than not only Buchberger's algorithm whose elimination of $\\t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2101.03482","kind":"arxiv","version":3},"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/2101.03482/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"}