{"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 $\\mu_d:=\\sum_{r=1}^d{(-1)^{r-1}}/{r!}$. 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 $d$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1511.07942","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"}