Waring's problem in algebraic number fields

Abstract

Let p be an odd prime and γ(k,pn) be the smallest positive integer s such that every integer is a sum of s k-th powers (modpn). We establish γ(k,pn)≤[k/2]+2 and γ(k,pn)≪k provided that k is not divisible by (p−1)/2. Next, let t=(p−1)/(p−1,k), and q be any positive integer. We show that if ϕ(t)≥q then γ(k,pn)≤c(q)k1/q for some constant c(q). These results generalize results known for the case of prime moduli. Next we generalize these results to a number field setting. Let F be a number field, R it's ring of integers and P a prime ideal in R that lies over a rational prime p with ramification index e, degree of inertia f and put t=(pf−1)/(p−1,k). Let k=prk1 with p∤k1 and γ(k,Pn) be the smallest integer s such that every algebraic integer in F that can be expressed as a sum of k-th powers (modPn) is expressible as a sum of s k-th powers (modPn). We prove for instance that when p>e+1 then γ(k,Pn)≤c(t)pnf/ϕ(t). Moreover, if p>e+1 we obtain the upper bounds \dsγ(k,Pn)≤2313(kk1)8.44/log⁡p+12 if f=2 or 3, and \dsγ(k,Pn)≤129(kk1)5.55/log⁡p+12 if f≥4. We also show that if P does not ramify then \dsγ(k,Pn)≤172(kk1)2.83/log⁡p+12 if f≥2 and k1≤pf/2, and \dsγ(k,Pn)≤(fpf/2−1)k if f>2 and k1>pf/2.