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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Minimal value set polynomials over fields of size $p^3$
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by Herivelto Borges and Lucas Reis PDF
Proc. Amer. Math. Soc. 149 (2021), 3639-3649 Request permission


For any prime number $p$, and integer $k\geqslant 1$, let $\mathbb {F}_{p^k}$ be the finite field of $p^k$ elements. A famous problem in the theory of polynomials over finite fields is the characterization of all nonconstant polynomials $F\in \mathbb {F}_{p^k}[x]$ for which the value set $\{F(\alpha ): \alpha \in \mathbb {F}_{p^k}\}$ has the minimum possible size $\left \lfloor (p^k-1)/\deg F \right \rfloor +1$. For $k\leqslant 2$, the problem was solved in the early 1960s by Carlitz, Lewis, Mills, and Straus. This paper solves the problem for $k=3$.
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Additional Information
  • Herivelto Borges
  • Affiliation: Universidade de São Paulo, Instituto de Ciências Matemáticas e de Computação, São Carlos, SP 13560-970, Brazil
  • MR Author ID: 857653
  • ORCID: 0000-0002-8100-3486
  • Email:
  • Lucas Reis
  • Affiliation: Departamento de Matemática, Universidade Federal de Minas Gerais, UFMG, Belo Horizonte, MG, 31270-901, Brazil
  • MR Author ID: 1170278
  • ORCID: 0000-0002-6224-9712
  • Email:
  • Received by editor(s): April 9, 2020
  • Received by editor(s) in revised form: December 23, 2020
  • Published electronically: June 4, 2021
  • Additional Notes: The second author was supported by FAPESP under grant 2018/03038-2
  • Communicated by: Rachel Pries
  • © Copyright 2021 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 149 (2021), 3639-3649
  • MSC (2020): Primary 11T06; Secondary 11G20
  • DOI:
  • MathSciNet review: 4291566