Sums of biquadrates and cubes in \(\mathbb F_q[t]\) (Q1430424)

The author shows that a polynomial \(P[t]\) over a field of \(q\) elements \((\text{gcd}(q,6)=1)\) can be represented for general \(q\) as the sum of 16 biquadrates and 7 cubes, with only 11 biquadrates if the sum is mixed (plus and minus). The method consists of first representing an arbitrary field element as such a sum [see \textit{R. Lidl} and \textit{H. Niederreiter}, Finite fields. Encycl. Math. Appl. 20, Cambridge: Cambridge University Press (1984; Zbl 0554.12010)]. Then \(P[t]\) is represented inductively as a sum of biquadrates (or cubes) of increasingly low degree, finally using identities for the lowest powers of \(t\) such as \[ 48t= (t+ 2)^4- 2(t+ 1)^4+ 2(t- 1)^4- (t- 2)^4. \] [See \textit{L. N. Vaserstein}, Waring's problem for algebras over fields, J. Number Theory 26, 286--298 (1987; Zbl 0624.10049).] A panply of special values of \(q\) is listed where the number of biquadrates (or cubes) is modified.