The Unrestricted Variant of Waring's Problem in Function Fields

Abstract

Let J k q [t] denote the additive closure of the set of k th powers in the polynomial ring Fq[t], defined over the finite field Fq having q elements. We show that when s>k + 1 and q>k 2k+2 , then every polynomial in J k q [t] is the sum of at most s k th powers of polynomials from Fq[t]. When k is large and s>( 4 3 + o(1))k log k , the same conclusion holds without restriction on q . Refinements are offered that depend on the characteristic of Fq .