{"paper":{"title":"Univariate Ideal Membership Parameterized by Rank, Degree, and Number of Generators","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","headline":"","cross_cats":[],"primary_cat":"cs.DS","authors_text":"Abhranil Chatterjee, Partha Mukhopadhyay, Rajit Datta, V. Arvind","submitted_at":"2018-08-31T14:38:53Z","abstract_excerpt":"Let $\\mathbb{F}[X]$ be the polynomial ring over the variables $X=\\{x_1,x_2, \\ldots, x_n\\}$. An ideal $I=\\langle p_1(x_1), \\ldots, p_n(x_n)\\rangle$ generated by univariate polynomials $\\{p_i(x_i)\\}_{i=1}^n$ is a \\emph{univariate ideal}. We study the ideal membership problem for the univariate ideals and show the following results.\n  \\item Let $f(X)\\in\\mathbb{F}[\\ell_1, \\ldots, \\ell_r]$ be a (low rank) polynomial given by an arithmetic circuit where $\\ell_i : 1\\leq i\\leq r$ are linear forms, and $I=\\langle p_1(x_1), \\ldots, p_n(x_n)\\rangle$ be a univariate ideal. Given $\\vec{\\alpha}\\in {\\mathbb{"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1808.10787","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"}