{"paper":{"title":"Represent MOD function by low degree polynomial with unbounded one-sided error","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.CC","authors_text":"Chris Beck, Yuan Li","submitted_at":"2013-04-02T17:50:54Z","abstract_excerpt":"In this paper, we prove tight lower bounds on the smallest degree of a nonzero polynomial in the ideal generated by $MOD_q$ or $\\neg MOD_q$ in the polynomial ring $F_p[x_1, \\ldots, x_n]/(x_1^2 = x_1, \\ldots, x_n^2 = x_n)$, $p,q$ are coprime, which is called \\emph{immunity} over $F_p$. The immunity of $MOD_q$ is lower bounded by $\\lfloor (n+1)/2 \\rfloor$, which is achievable when $n$ is a multiple of $2q$; the immunity of $\\neg MOD_q$ is exactly $\\lfloor (n+q-1)/q \\rfloor$ for every $q$ and $n$. Our result improves the previous bound $\\lfloor \\frac{n}{2(q-1)} \\rfloor$ by Green.\n  We observe how"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1304.0713","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"}