Every polynomial over a field containing $\mathbb F_{16}$ is a strict sum of four cubes and one expression $A^2+A$
Bull. Korean Math. Soc. 2009 Vol. 46, No. 5, 941-947
Printed September 1, 2009
Luis H. Gallardo
University of Brest
Abstract : Let $q$ be a power of $16.$ Every polynomial $P \in {\mathbb F}_q[t]$ is a strict sum $$ P = A^2+A + B^3+C^3+D^3+E^3. $$ The values of $A,B,C,D,E$ are effectively obtained from the coefficients of $P.$ The proof uses the new result that every polynomial $Q \in {\mathbb F}_q[t],$ satisfying the necessary condition that the constant term $Q(0)$ has zero trace, has a strict and effective representation as: $$ Q = F^2 + F + tG^2. $$ This improves for such $q$'s and such $Q$'s a result of Gallardo, Rahavandrainy, and Vaserstein that requires three polynomials $F,G,H$ for the strict representation $Q = F^2+F +GH.$ Observe that the latter representation may be considered as an analogue in characteristic $2$ of the strict representation of a polynomial $Q$ by three squares in odd characteristic.
Keywords : Waring's problem, quadratic polynomials, cubes, finite fields, characteristic $2$
MSC numbers : 11T55, 11T06
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