Cochrane, Todd E.Hart, DerrickPinner, Christopher G.Spencer, Craig2014-11-252014-11-252014-11-25http://hdl.handle.net/2097/18747Let p be a prime, Z_p be the finite field in p elements, k be a positive integer, and A be the multiplicative subgroup of nonzero k-th powers in Z_p. The goal of this paper is to determine, for a given positive integer s, a value t_s such that if |A| ≫ t_s then every element of Z_p is a sum of s k-th powers. We obtain t_4 = p^{\frac{22}{39} + \in}, t_5 = p^{\frac{15}{29} + \in} and for s s ≥ 6, t_s = p^{\frac{9s+45}{29s+33} + \in}. For s ≥ 24 further improvements are made, such as t_32 = p^{\frac{5}{16} + \in} and t_128 = p^{\frac{1}{4}}.en-USThis Item is protected by copyright and/or related rights. You are free to use this Item in any way that is permitted by the copyright and related rights legislation that applies to your use. For other uses you need to obtain permission from the rights-holder(s).http://rightsstatements.org/vocab/InC/1.0/Warings problemExponential sumsSum-Product setsArticle (author version)