{
  "abstract": "We give a complete classification of minimal value set polynomials (MVSPs), settling a problem introduced by Carlitz, Lewis, Mills and Straus in 1961. For MVSPs $F \\in \\mathbb{F}_q[x]$ with more than two values, earlier work showed that their value sets $\\left\\{F(a): a \\in \\mathbb{F}_q\\right\\}$ can be reduced to the form $\\mathcal U^v=\\{u^v:u\\in \\mathcal U\\}$, where $\\mathcal U$ is a vector space over a subfield of $\\mathbb F_q$, and $v\\geq 1$ is a suitable integer. This classification proceeds through two parallel constructions, according to whether $\\mathcal U$ is itself a field or not, governed respectively by the cyclic multiplicative structure and by the additive linearized structure. In both cases, Hasse derivatives impose constraints on the root multiplicities of a distinguished factor of $F$, leading to a factorization that preserves the underlying structure in a polynomial of smaller degree and thus allows the argument to be iterated.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.37877",
  "author_total": 2,
  "authors": [
    {
      "name": "Herivelto Borges"
    },
    {
      "name": "Lucas Reis"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1sgtjdszh84d5.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.37877",
  "fulltext_url": null,
  "id": "1sgtjdszh84d5",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.37877",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.NT"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.37877",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1sgtjdszh84d5.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2610.37877",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.37877"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "A complete classification of minimal value set polynomials",
  "title_text": "A complete classification of minimal value set polynomials",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/1sgtjdszh84d5",
  "venue": "arXiv",
  "versions": []
}