Waring's problem in function fields

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.

Locations

• UWSpace (University of Waterloo) - View - PDF
• Journal für die reine und angewandte Mathematik (Crelles Journal) - View