{"paper":{"title":"Numerical Decomposition of Affine Algebraic Varieties","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Gerhard Pfister, Shawki Al-Rashed","submitted_at":"2010-10-15T11:33:10Z","abstract_excerpt":"An irreducible algebraic decomposition $\\cup_{i=0}^{d}X_i=\\cup_{i=0}^{d} (\\cup_{j=1}^{d_i}X_{ij})$ of an affine algebraic variety X can be represented as an union of finite disjoint sets $\\cup_{i=0}^{d}W_i=\\cup_{i=0} ^{d}(\\cup_{j=1}^{d_i}W_{ij})$ called numerical irreducible decomposition (cf. [14],[15],[17],[18],[19],[21],[22],[23]). $W_i$ corresponds to a pure i-dimensional $X_i$, and $W_{ij}$ presents an i- dimensional irreducible component $X_{ij}$. Modifying this concepts by using partially Gr\\\"obner bases, local dimension, and the \"Zero Sum Relation\" we present in this paper an implement"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1010.3129","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"}