Waring's problem in function fields

Abstract

Let Fq½t denote the ring of polynomials over the finite field Fq of characteristic p, and write Jk q ½t for the additive closure of the set of kth powers of polynomials in Fq½t. Define GqðkÞ to be the least integer s satisfying the property that every polynomial in Jk q ½t of su‰ciently 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 GqðkÞeCk log k þ Oðk log log kÞ. Here, the coe‰cient 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 e4=3. There are associated conclusions for the solubility of diagonal equations over Fq½t, and for exceptional set estimates in Waring’s problem.