PORTUGALIAE MATHEMATICA Vol. 61, No. 1, pp. 3549 (2004) 

Waring's problem for polynomial cubes and squares over a finite field with odd characteristicLuis GallardoDepartment of Mathematics, University of Brest,6, Avenue Le Gorgeu, C.S.\ 93837, 29238 Brest Cedex 3  FRANCE Email: Luis.Gallardo@univbrest.fr 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 nonnegative 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: Keywords: Waring's problem; polynomials; finite fields. Classification (MSC2000): 11T55, 11P05, 11D85. Full text of the article:
Electronic version published on: 7 Mar 2008.
© 2004 Sociedade Portuguesa de Matemática
