A complete classification of minimal value set polynomials

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 ∈ 𝔽_q[x] with more than two values, earlier work showed that their value sets {F(a): a ∈ 𝔽_q} can be reduced to the form 𝒰^v={u^v:u∈ 𝒰}, where 𝒰 is a vector space over a subfield of 𝔽_q, and v≥ 1 is a suitable integer. This classification proceeds through two parallel constructions, according to whether 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.

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BibTeX RIS

BibTeX

@misc{borges2026complete,
  title = {{A complete classification of minimal value set polynomials}},
  author = {Herivelto Borges and Lucas Reis},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.37877},
  eprint = {2610.37877},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.37877},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - A complete classification of minimal value set polynomials
AU  - Herivelto Borges
AU  - Lucas Reis
PY  - 2026
DA  - 2026/10/27
PB  - arXiv
DO  - 10.48550/arxiv.2610.37877
UR  - https://arxiv.org/abs/2610.37877
AB  - 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 ∈ 𝔽_q[x] with more than two values, earlier work showed that their value sets {F(a): a ∈ 𝔽_q} can be reduced to the form 𝒰^v={u^v:u∈ 𝒰}, where 𝒰 is a vector space over a subfield of 𝔽_q, and v≥ 1 is a suitable integer. This classification proceeds through two parallel constructions, according to whether 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.
ER  -

Abstract

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