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We establish conditions on the family $\\mathcal{A}$ under which $\\mathcal{V}(\\mathcal{A})=\\mu_d\\,q+\\mathcal{O}(q^{1/2})$, where $\\mu_d:=\\sum_{r=1}^d{(-1)^{r-1}}/{r!}$. The result holds without any restriction on the characteristic of $\\mathbb{F}_{\\hskip-0.7mm q}$ and provides an explicit expression for the constant underlying the $\\mathcal{O}$--notation in terms of $d$"},"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":"1511.07942","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-11-25T03:26:54Z","cross_cats_sorted":[],"title_canon_sha256":"a00450b40b4571ef0be440b57f4ead147002bfa30d45fa051781c526b8039fc8","abstract_canon_sha256":"2814be8bc7ed5ee29036864afc4dfbb471548286b2b0c653d0ac42ccc8f037b5"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:26:02.147979Z","signature_b64":"qWYJIYsd3qLLY4JHhwT3jryVOVIx2mimqPqARpZ0M3ZSHW496fGd29ZjGKJu+UCCXLihCGLekKD9f5zwIQ5TAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"efe9de2b5c92397072129edf22ca13a00d2796ce0a50c058092b5f2afae6506a","last_reissued_at":"2026-05-18T01:26:02.147422Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:26:02.147422Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the value set of small families of polynomials over a finite field, III","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Guillermo Matera, Mariana P\\'erez, Melina Privitelli","submitted_at":"2015-11-25T03:26:54Z","abstract_excerpt":"We estimate the average cardinality $\\mathcal{V}(\\mathcal{A})$ of the value set of a general family $\\mathcal{A}$ of monic univariate polynomials of degree $d$ with coefficients in the finite field $\\mathbb{F}_{\\hskip-0.7mm q}$. We establish conditions on the family $\\mathcal{A}$ under which $\\mathcal{V}(\\mathcal{A})=\\mu_d\\,q+\\mathcal{O}(q^{1/2})$, where $\\mu_d:=\\sum_{r=1}^d{(-1)^{r-1}}/{r!}$. 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