Exact solutions to Waring’s problem for finite fields

with xi ∈ Fq, i.e., as a sum of kth powers of elements of Fq. We can then define the Waring function g(k, q) as the maximal number of summands needed to express all elements of Fq as sums of kth powers. We note that, by an easy argument, we have g(k, q) = g(k, q), where k = gcd(k, q − 1). Hence, we will assume from now on that k divides q − 1. Several authors have established bounds on the value of g(k, q) for various choices of the parameters k and q – a survey is given in [8]. For the cases where the exponent k is small compared to q, there are strong results. For example, whenever 2 ≤ k < q + 1, it follows that g(k, q) = 2 by a direct application of the Weil bound for the number of points on varieties over finite fields [6, 7, 8]. In this paper, we will look at the cases where the exponent k is large compared to q, and we will obtain not only a bound, but the exact value of g(k, q) for two infinite families of pairs (k, q). Our main results are the following.