Abstract
Let 
denote the ring of polynomials over the finite field 𝕗q of characteristic p, and write 
for the additive closure of the set of kth powers of polynomials in . Define Gq(k) to be the least integer s satisfying the property that every polynomial in of sufficiently large degree admits a strict representation as a sum of skth powers. We employ a version of the Hardy-Littlewood method involving the use of smooth polynomials in order to establish a bound of the shape Gq(k) ≦ Ck log k + O(k log log k). Here, the coefficient C is equal to 1 when k < p, and C is given explicitly in terms of k and p when k > p, but in any case satisfies C ≦ 4/3. There are associated conclusions for the solubility of diagonal equations over , and for exceptional set estimates in Waring's problem.
© Walter de Gruyter Berlin · New York 2010
