{"paper":{"title":"Polynomials Meeting Ax's Bound","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Xiang-dong Hou","submitted_at":"2015-12-15T22:38:27Z","abstract_excerpt":"Let $f\\in\\Bbb F_q[X_1,\\dots,X_n]$ with $\\deg f=d>0$ and let $Z(f)=\\{(x_1,\\dots,x_n)\\in \\Bbb F_q^n: f(x_1,\\dots,x_n)=0\\}$. Ax's theorem states that $|Z(f)|\\equiv 0\\pmod {q^{\\lceil n/d\\rceil-1}}$, that is, $\\nu_p(|Z(f)|)\\ge m(\\lceil n/d\\rceil-1)$, where $p=\\text{char}\\,\\Bbb F_q$, $q=p^m$, and $\\nu_p$ is the $p$-adic valuation. In this paper, we determine a condition on the coefficients of $f$ that is necessary and sufficient for $f$ to meet Ax's bound, that is, $\\nu_p(|Z(f)|)=m(\\lceil n/d\\rceil-1)$. Let $R_q(d,n)$ denote the $q$-ary Reed-Muller code $\\{f\\in\\Bbb F_q[X_1,\\dots,X_n]: \\deg f\\le d,\\ "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1512.04997","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"}