Sums and strict sums of biquadrates in F_q[t],qε{3,9}

Let q be a power of a prime number. Observe that just for q ε {3,9} some congruence obstructions occur to the representation of polynomials in F_q[t] as a sum (and so also as a strict sum) of biquadrates. We define g(4,F_q[t}) as the least g such that every polynomial that is a strict sum of biquadrates is a strict sum of g biquadrates. We compare the set of sums of biquadrates with the set of strict sums of biquadrates for q ε {3,9}. Our main result is that g(4,F_q[t]) ≤14 when q ε {3,9}. The set of sums of cubes in F₄[t] is also determined. This completes the study of the case of representation by sums of cubes (in which the congruence obstructions occur only for 9 ε{2,4}).