{"paper":{"title":"A Note on the Chevalley--Warning Theorems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG"],"primary_cat":"math.NT","authors_text":"D.R. Heath-Brown","submitted_at":"2010-09-20T10:41:28Z","abstract_excerpt":"Let $f_1,\\...,f_r$ be polynomials in $n$ variables over a finite field $F$ of cardinality $q$ and characteristic $p$. Let $f_i$ have total degree $d_i$ and define $d=d_1+\\...+d_r$. Write $Z$ for the set of common zeros of the $f_i$, over the field $F$. Warning showed that $#(Z\\cap H_1)\\equiv#(Z\\cap H_2)\\mod{p}$ for any two parallel affine hyperplanes $H_1,H_2$ in $F^n$. We prove that the same congruence holds to modulus $q$. Warning also proved that $# Z\\ge q^{n-d}$ providing that $Z$ is non-empty. We sharpen this inequality in various ways, assuming that $Z$ is not a linear subspace of $F^n$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1009.3764","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"}