Waring's number mod m

Abstract

Let p be an odd prime and γ (k, p n ) be the smallest positive integer s such that every integer is a sum of s kth powers (mod p n ). We establish γ (k, p n ) [k/2] + 2 and γ (k, p n ) √ 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, p n ) c(q)k 1/q for some constant c(q). These results generalize results known for the case of prime moduli. Video abstract: For a video summary of this paper, please visit http://www.youtube.com/watch?v= zpHYhwL1kD0.

On the Odlyzko–Stanley enumeration problem and Waringʼs problem over finite fields

“…For instance, Bourgain and Garaev proved in [3] that if δ > 1/4, then one can take δ ′ = 0.000015927 + o (1). Taking H = {x m , x ∈ F * p } and following the same argument of Konyagin and Shparlinski [19], the above bound immediately implies that if m < p 1−δ , then…”

“…This work is supported by the National Science Foundation of China (11001170). 1 Note that N * m (b) = q k=0 N * m (k, b) and it is sufficient to consider the case |S| ≤ (q − 1)/2 by symmetry. Hence (1.2) and (1.3) follow from (1.4) directly.…”

“…Note that we can always assume that m < (p − 1)/2. The first bound γ(m, p) ≤ m for any prime p was proved by Cauchy in 1813, as reported in [1]. Dozens of papers, for instance, [11,12,13,14,15,17,18,6,24], studied Waring's number mod a prime number, and generally, Warng's number mod an integer, Waring's number over finite fields, p-adic integers and a general commutative ring.…”

On Waring numbers of henselian rings

“…To be precise, we ask for determining real or complex numbers such that the existence of their embeddings in R implies the equality w n (R) = w n (K), without computing w n (R) or w n (K). In case of n = 2, which motivates the above problem, [14,Theorem 4.5] proved by Kneser and Colliot-Thélène states that 1 2 ∈ R implies w 2 (R) = w 2 (K). For n = 3 one has to expect that some nonrational numbers have to be involved as the following example of complete discrete valuation ring of equicharacteristic zero shows.…”

“…where G(n) is the function related to the Waring problem. See also [1,37,41,42] for Waring problem in finite rings, [6,36] for number fields, [7,8,46] for p-adic rings and [31,33,39,44,45] for more general rings. There is an interesting algorithmic approach to the problem of sum of squares [17,18,27,28].…”