Let Fq[t] denote the ring of polynomials over the finite field q of characteristic p, and write Jq^k[t] for the additive closure of the set of kth powers of polynomials in Fq[t]. Define Gq(k) to be the least integer s satisfying the property that every polynomial in Fq[t] of sufficiently large degree admits a strict representation as a sum of s kth 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 Fq[t], and for exceptional set estimates in Waring's problem.
|Translated title of the contribution||Waring's problem in function fields|
|Pages (from-to)||1 - 67|
|Number of pages||67|
|Journal||Journal für die reine und angewandte Mathematik|
|Early online date||23 Nov 2009|
|Publication status||Published - 2010|