{"paper":{"title":"On value sets of polynomials over a field","license":"","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Zhi-Wei Sun","submitted_at":"2007-03-07T17:26:21Z","abstract_excerpt":"Let F be any field. Let p(F) be the characteristic of F if F is not of characteristic zero, and let p(F)=+\\infty otherwise. Let A_1,...,A_n be finite nonempty subsets of F, and let\n  $$f(x_1,...,x_n)=a_1x_1^k+...+a_nx_n^k+g(x_1,...,x_n)\\in F[x_1,...,x_n]$$ with k in {1,2,3,...}, a_1,...,a_n in F\\{0} and deg(g)<k. We show that $$|{f(x_1,...,x_n):x_1 in A_1,...,x_n in A_n}| \\geq min{p(F),\\sum_{i=1}^n[(|A_i|-1)/k]+1}.$$ When $k\\geq n$ and $|A_i|\\geq i$ for $i=1,...,n$, we also have $$|{f(x_1,...,x_n):x_1 in A_1,...,x_n in A_n, and x_i not=x_j if i not=j}| \\geq min{p(F),\\sum_{i=1}^n[(|A_i|-i)/k]+1"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0703180","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/math/0703180/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}