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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,\\ "},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1512.04997","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-12-15T22:38:27Z","cross_cats_sorted":[],"title_canon_sha256":"83b3c444068e357928c43bf49ab0d39386f2bd4fb7d260964b5210278fa5fc52","abstract_canon_sha256":"ae39620c98eb71ca6c2013ef274630cd39880c93dc49baa60eb2f38221c38b01"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:24:13.771157Z","signature_b64":"0xCTNRscOlXi22EbOBrkGirwyE9d9maVI2+ijd03/O2O2KJq8XwI7ZQHcOgLuuHlreJsyYOZTlbNPlWSIj6EAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a5213598960ab3723ca6cc2e1cf662954bb1ce262e1e0be6ba4ef90bd7846b75","last_reissued_at":"2026-05-18T01:24:13.770748Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:24:13.770748Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1512.04997","created_at":"2026-05-18T01:24:13.770812+00:00"},{"alias_kind":"arxiv_version","alias_value":"1512.04997v1","created_at":"2026-05-18T01:24:13.770812+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1512.04997","created_at":"2026-05-18T01:24:13.770812+00:00"},{"alias_kind":"pith_short_12","alias_value":"UUQTLGEWBKZX","created_at":"2026-05-18T12:29:44.643036+00:00"},{"alias_kind":"pith_short_16","alias_value":"UUQTLGEWBKZXEPFG","created_at":"2026-05-18T12:29:44.643036+00:00"},{"alias_kind":"pith_short_8","alias_value":"UUQTLGEW","created_at":"2026-05-18T12:29:44.643036+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/UUQTLGEWBKZXEPFGZQXBZ5TCSV","json":"https://pith.science/pith/UUQTLGEWBKZXEPFGZQXBZ5TCSV.json","graph_json":"https://pith.science/api/pith-number/UUQTLGEWBKZXEPFGZQXBZ5TCSV/graph.json","events_json":"https://pith.science/api/pith-number/UUQTLGEWBKZXEPFGZQXBZ5TCSV/events.json","paper":"https://pith.science/paper/UUQTLGEW"},"agent_actions":{"view_html":"https://pith.science/pith/UUQTLGEWBKZXEPFGZQXBZ5TCSV","download_json":"https://pith.science/pith/UUQTLGEWBKZXEPFGZQXBZ5TCSV.json","view_paper":"https://pith.science/paper/UUQTLGEW","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1512.04997&json=true","fetch_graph":"https://pith.science/api/pith-number/UUQTLGEWBKZXEPFGZQXBZ5TCSV/graph.json","fetch_events":"https://pith.science/api/pith-number/UUQTLGEWBKZXEPFGZQXBZ5TCSV/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/UUQTLGEWBKZXEPFGZQXBZ5TCSV/action/timestamp_anchor","attest_storage":"https://pith.science/pith/UUQTLGEWBKZXEPFGZQXBZ5TCSV/action/storage_attestation","attest_author":"https://pith.science/pith/UUQTLGEWBKZXEPFGZQXBZ5TCSV/action/author_attestation","sign_citation":"https://pith.science/pith/UUQTLGEWBKZXEPFGZQXBZ5TCSV/action/citation_signature","submit_replication":"https://pith.science/pith/UUQTLGEWBKZXEPFGZQXBZ5TCSV/action/replication_record"}},"created_at":"2026-05-18T01:24:13.770812+00:00","updated_at":"2026-05-18T01:24:13.770812+00:00"}