THE UNRESTRICTED VARIANT OF WARING'S PROBLEM IN FUNCTION FIELDS

Let J(q)(k)[t] denote the additive closure of the set of kth powers in the polynomial ring IF q[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(q)(k)[t] is the sum of at most s kth powers of polynomials from F-q[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 F-q.