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.
Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callMore From: Journal für die reine und angewandte Mathematik (Crelles Journal)
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
