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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 $F^n$."},"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":"1009.3764","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2010-09-20T10:41:28Z","cross_cats_sorted":["math.AG"],"title_canon_sha256":"38c5b56592f97ea4060fe1a5fa5bc3f0ab0aaf5703abd9a911858ae22bb77e08","abstract_canon_sha256":"fa07fcae3f41ab887b478ed725c5d455db3c7859a8253b631d8b4b0a23c762ac"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:04:01.410718Z","signature_b64":"0h2oLagQyiJ9O4LoHpg1WieCGyPdAd/0A/euqdefqYPi6CIhqWD3Y46bvKrHwsdPKFb/Ar0cIFzX573HZmL7Bg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d8fe35aec7b3a44b57557c51e03dcc12ec8bf36fed28aa72595979ed6a890f9b","last_reissued_at":"2026-05-18T01:04:01.410124Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:04:01.410124Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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. 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