{"paper":{"title":"An epsilon-delta bound for plane algebraic curves and its use for certified homotopy continuation of systems of plane algebraic curves","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG"],"primary_cat":"math.CV","authors_text":"Stefan Kranich","submitted_at":"2015-05-13T15:40:22Z","abstract_excerpt":"We explain how, given a plane algebraic curve $\\mathcal{C}\\colon f(x,y) = 0$, $x_1 \\in \\mathbb{C}$ not a singularity of $y$ w.r.t. $x$, and $\\varepsilon > 0$, we can compute $\\delta > 0$ such that $|y_j(x_1) - y_j(x_2)| < \\varepsilon$ for all holomorphic functions $y_j(x)$ which satisfy $f(x, y_j(x)) = 0$ in a neighbourhood of $x_1$ and for all $x_2$ with $|x_1 - x_2| < \\delta$. Consequently, we obtain an algorithm for reliable homotopy continuation of plane algebraic curves. As an example application, we study continuous deformation of closed discrete Darboux transforms.\n  Moreover, we discus"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1505.03432","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"}