{"paper":{"title":"Bounding the degrees of a minimal $\\mu$-basis for a rational surface parametrization","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Yairon Cid-Ruiz","submitted_at":"2016-11-22T20:43:12Z","abstract_excerpt":"In this paper, we study how the degrees of the elements in a minimal $\\mu$-basis of a parametrized surface behave. For an arbitrary rational surface parametrization $P(s,t)=(a_1(s,t),a_2(s,t),a_3(s,t),a_4(s,t)) \\in \\mathbb{F}[s,t]^4$ over an infinite field $\\mathbb{F}$, we show the existence of a $\\mu$-basis with polynomials bounded in degree by $O(d^{33})$, where $d=\\max(\\text{deg}(a_1),\\text{deg}(a_2), \\text{deg}(a_3), \\text{deg}(a_4))$. Under additional assumptions we can obtain tighter bounds."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1611.07506","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":""},"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"}