{"paper":{"title":"On the Complexity of Computing with Planar Algebraic Curves","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.NA","math.AG","math.GT","math.NA"],"primary_cat":"cs.SC","authors_text":"Alexander Kobel, Michael Sagraloff","submitted_at":"2014-01-22T14:54:13Z","abstract_excerpt":"In this paper, we give improved bounds for the computational complexity of computing with planar algebraic curves. More specifically, for arbitrary coprime polynomials $f$, $g \\in \\mathbb{Z}[x,y]$ and an arbitrary polynomial $h \\in \\mathbb{Z}[x,y]$, each of total degree less than $n$ and with integer coefficients of absolute value less than $2^\\tau$, we show that each of the following problems can be solved in a deterministic way with a number of bit operations bounded by $\\tilde{O}(n^6+n^5\\tau)$, where we ignore polylogarithmic factors in $n$ and $\\tau$:\n  (1) The computation of isolating reg"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1401.5690","kind":"arxiv","version":2},"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"}