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

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}