Bulletin of the Belgian Mathematical Society - Simon Stevin

Waring's problem for cubes and squares over a finite field of even characteristic

Luis Gallardo

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Abstract

Let $q$ be a power of a prime $p \neq 3.$ We characterize the following two sets of polynomials: $M(q)=\{P \in {\bf F}_{q}[t]$ such that $P$ is a strict sum of cubes in ${\bf F}_{q}[t]\}$ and $S(q)=\{P \in {\bf F}_{q}[t]$ such that $P$ is a strict sum of cubes and squares in ${\bf F}_{q}[t]\}.$ Let $g(3,{\bf F}_{q}[t])=g \geq 0$ be the minimal integer such that every $P \in M(q)$ is a strict sum of $g$ cubes. Similarly let $g_1(3,2,{\bf F}_{q}[t])=g$ be the minimal integer such that every $P \in S(q)$ is a strict sum of $g$ cubes and a square. Our main result is:\begin{itemize} \item[i)] $4 \leq g(3,{\bf F}_{q}[t]) \leq 9\,\,\,$ for $q \in \{2,4\}.$ \item[ii)] $3 \leq g_1(3,2,{\bf F}_{q}[t]) \leq 4\,\,\,$ for $q =4.$ \end{itemize}

Article information

Source
Bull. Belg. Math. Soc. Simon Stevin, Volume 12, Number 3 (2005), 349-362.

Dates
First available in Project Euclid: 8 September 2005

Permanent link to this document
https://projecteuclid.org/euclid.bbms/1126195340

Digital Object Identifier
doi:10.36045/bbms/1126195340

Mathematical Reviews number (MathSciNet)
MR2173698

Zentralblatt MATH identifier
1111.11059

Subjects
Primary: 11T55: Arithmetic theory of polynomial rings over finite fields 11P05: Waring's problem and variants 11D85: Representation problems [See also 11P55]

Keywords
Waring's Problem Polynomials Finite Fields Characteristic 2

Citation

Gallardo, Luis. Waring's problem for cubes and squares over a finite field of even characteristic. Bull. Belg. Math. Soc. Simon Stevin 12 (2005), no. 3, 349--362. doi:10.36045/bbms/1126195340. https://projecteuclid.org/euclid.bbms/1126195340


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