## Minimal value set polynomials over fields of size $p^3$

HTML articles powered by AMS MathViewer

- by Herivelto Borges and Lucas Reis PDF
- Proc. Amer. Math. Soc.
**149**(2021), 3639-3649 Request permission

## Abstract:

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 $k=3$.## References

- Herivelto Borges,
*Frobenius nonclassical components of curves with separated variables*, J. Number Theory**159**(2016), 402–425. MR**3412730**, DOI 10.1016/j.jnt.2015.07.006 - Herivelto Borges and Ricardo Conceição,
*A new family of Castle and Frobenius nonclassical curves*, J. Pure Appl. Algebra**222**(2018), no. 4, 994–1002. MR**3720864**, DOI 10.1016/j.jpaa.2017.06.002 - Herivelto Borges and Ricardo Conceição,
*On the characterization of minimal value set polynomials*, J. Number Theory**133**(2013), no. 6, 2021–2035. MR**3027951**, DOI 10.1016/j.jnt.2012.08.030 - H. Borges, A. Sepúlveda, and G. Tizziotti,
*Weierstrass semigroup and automorphism group of the curves $\mathcal {X}_{n,r}$*, Finite Fields Appl.**36**(2015), 121–132. MR**3396379**, DOI 10.1016/j.ffa.2015.07.004 - L. Carlitz, D. J. Lewis, W. H. Mills, and E. G. Straus,
*Polynomials over finite fields with minimal value sets*, Mathematika**8**(1961), 121–130. MR**139606**, DOI 10.1112/S0025579300002230 - Arnaldo García,
*The curves $y^n=f(x)$ over finite fields*, Arch. Math. (Basel)**54**(1990), no. 1, 36–44. MR**1029595**, DOI 10.1007/BF01190666 - Javier Gomez-Calderon,
*A note on polynomials with minimal value set over finite fields*, Mathematika**35**(1988), no. 1, 144–148. MR**962743**, DOI 10.1112/S0025579300006355 - Javier Gomez-Calderon and Daniel J. Madden,
*Polynomials with small value set over finite fields*, J. Number Theory**28**(1988), no. 2, 167–188. MR**927658**, DOI 10.1016/0022-314X(88)90064-9 - J. W. P. Hirschfeld, G. Korchmáros, and F. Torres,
*Algebraic curves over a finite field*, Princeton Series in Applied Mathematics, Princeton University Press, Princeton, NJ, 2008. MR**2386879** - W. H. Mills,
*Polynomials with minimal value sets*, Pacific J. Math.**14**(1964), 225–241. MR**159813** - Gary L. Mullen (ed.),
*Handbook of finite fields*, Discrete Mathematics and its Applications (Boca Raton), CRC Press, Boca Raton, FL, 2013. MR**3087321**, DOI 10.1201/b15006 - Rudolf Lidl and Harald Niederreiter,
*Finite fields*, 2nd ed., Encyclopedia of Mathematics and its Applications, vol. 20, Cambridge University Press, Cambridge, 1997. With a foreword by P. M. Cohn. MR**1429394** - Karl-Otto Stöhr and José Felipe Voloch,
*Weierstrass points and curves over finite fields*, Proc. London Math. Soc. (3)**52**(1986), no. 1, 1–19. MR**812443**, DOI 10.1112/plms/s3-52.1.1 - Da Qing Wan, Peter Jau-Shyong Shiue, and Ching Shyang Chen,
*Value sets of polynomials over finite fields*, Proc. Amer. Math. Soc.**119**(1993), no. 3, 711–717. MR**1155603**, DOI 10.1090/S0002-9939-1993-1155603-2

## 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: hborges@icmc.usp.br
**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: lucasreismat@mat.ufmg.br
- 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: https://doi.org/10.1090/proc/15478
- MathSciNet review: 4291566