On the p - adic Waring ’ s problem

Let R be a ring (commutative with unity, in what follows). For an integer n > 1 define gR(n) to be the least integer s for which every element of R is a sum of s nth powers of elements of R, if such an integer exists, or ∞ otherwise. Waring’s problem for R is the problem of deciding whether gR(n) is finite and estimating it for all n. Note that what is usually called Waring’s problem is not what we call Waring’s problem for Z. For n odd, what we call Waring’s problem for Z is usually referred to as the “easier” Waring’s problem, with Waring’s problem proper referring only to positive integers. Nevertheless, the results we are discussing here have an impact on the usual Waring’s problem because they have a bearing on the issue of local solvability. For Waring’s problem for finite fields see [GV] and the references therein. We wish to consider in this note Waring’s problem for unramified extensions of the ring of p-adic integers Zp. For Zp the problem has been considered extensively (see [B] and references therein) for its connection with the problem of non-vanishing of the singular series in the classical Waring’s problem. We shall improve some of Bovey’s results for Zp and obtain new results for unramified extensions of Zp. Let W (k) be the (unique) complete unramified extension of Zp with residue field k algebraic over Fp; W (k) is the ring of Witt vectors over k and we will recall some of its properties later. To begin with, note that it follows from Hensel’s lemma that if n = pd, (p, d) = 1 and a ≡ x1 + . . .+xs (mod pt+ε), x1, . . . , xs ∈ W (k), where ε = 1, p 6= 2, ε = 2, p = 2 and some xi is a unit, then there exist y1, . . . , ys ∈W (k) with a = y 1 + . . .+ y s . This is easy and well known. Assume for now on that p 6= 2 so ε = 1. Notice that if a, as above, is a unit then, for any representation a ≡ x1 + . . .+xs (mod pt+ε), some xi will be a unit. So every unit of W (k) is a sum of at most gWt+1(k)(n) nth powers,