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In this paper we consider the case when $X= \\mathbb X (k)$ where $\\mathbb X$ is a complete intersection of bounded codimension defined by a high rank polynomials of degrees $d, char(k)=0$ or $char (k)>d$ and either $k$ is algebraically closed, or $k=\\mathbb F _q,q>ad$. We show that under these assumptions any $k$-valued weakly polynomial function of degree $ \\leq a"},"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":"1808.09439","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2018-08-28T17:54:48Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"21bc9a8f37c64dc1a09240b8a526141fff757a0471007389af56d3941ede60f6","abstract_canon_sha256":"67a577e759a04cb50fa018698ab2ba5939d066436cf2586811db869adeb18474"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:54:48.229508Z","signature_b64":"m+19Bib3mgjsO71PZXkUeCmNfuxPcQCsmSeAdcBX2Ig/QkYY1qDmysTh9PzzTHrsrskXjKkbdxpXnJQzMIkZCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4fed6703ee4df542ff71e579bac369263e83f36b443c582ebb8c297b21c5b7c7","last_reissued_at":"2026-05-17T23:54:48.228850Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:54:48.228850Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Extending weakly polynomial functions from high rank varieties","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.AG","authors_text":"David Kazhdan, Tamar Ziegler","submitted_at":"2018-08-28T17:54:48Z","abstract_excerpt":"Let $k$ be a field, $V$ a $k$-vector space and $X$ be a subset of $V $. A function $f:X\\to k$ is weakly polynomial of degree $\\leq a$, if the restriction of $f$ on any affine subspace $L\\subset X$ is a polynomial of degree $\\leq a$. In this paper we consider the case when $X= \\mathbb X (k)$ where $\\mathbb X$ is a complete intersection of bounded codimension defined by a high rank polynomials of degrees $d, char(k)=0$ or $char (k)>d$ and either $k$ is algebraically closed, or $k=\\mathbb F _q,q>ad$. 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