Waring's problem for polynomial cubes and squares over a finite field with odd characteristic

Luis Gallardo

Abstract: Let $q$ be a power of an odd prime $p$. For $r\in\{1,2\}$ and $p\neq 3$, we give bounds for the minimal non-negative integer $g\sb{r}(3,2,{\bf F}\sb{q}[t])=g$, such that every $P\in{\bf F}\sb{q}[t]$ is a strict sum of $g$ cubes and $r$ squares. Similarly we study for $p=3$, the number $g\sb{1}(2,3,{\bf F}\sb{q}[t])=g$, such that every $P\in{\bf F}\sb{q}[t]$ is a strict sum of $g$ squares and a cube. All bounds are obtained using explicit representations. Precisely our main results are:
(i) $2\leq g\sb{1}(3,2,{\bf F}\sb{q}[t])\leq 5$ when $p\neq 3$ and $q\notin\{7,13\}$.
(ii) $1\leq g\sb{2}(3,2,{\bf F}_{q}[t])\leq 4$ when $p\neq 3$ and for all $q\neq 7$.
(iii) $2\leq g\sb{1}(2,3,{\bf F}\sb{q}[t])\leq 3$ for all $q$ when $p=3$.
The later item is of some interest since Serre gave an indirect proof of the fact that for $q\neq 3$ every polynomial in ${\bf F}\sb{q}[t]$ is a strict sum of $3$ squares, and that for $q=3$ there are some exceptions ($8$ as precised by Webb) that require $4$ squares.

Keywords: Waring's problem; polynomials; finite fields.

Classification (MSC2000): 11T55, 11P05, 11D85.

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