A note on Waring's problem in finite fields

(1) g(k, p) exists if and only if p − 1 pd − 1 k for all d |n, d 6= n. We shall suppose from now on that g(k, p) exists. Several bounds for g(k, p) are known. For surveys see [7] and [13]. Recent results can be found in [5]–[9] and [13]. In the case n = 1 it was proved in [4, Theorem 1] that (2) g(k, p) < 68k(ln k) for k < (p− 1)/2. Whether (2) holds true for n > 1 has not been known yet. In this note we prove g(k, p) < 6.2n(2k) ln k, which yields an extension of (2) to arbitrary n. Moreover, we show g(k, p) > 2 (((n+ 1)k) 1/n − 1) if n + 1 is a prime such that p is a primitive root modulo n + 1 and k = (p − 1)/(n+ 1).