WARING'S PROBLEM WITH POLYNOMIAL SUMMANDS

Abstract

Let f(x) be an integer-valued polynomial with no fixed integer divisor [ges ] 2, that is, for no integer d [ges ] 2 does d[mid ]f(x) for all integers x. One generalization of the famous Waring problem is to determine whether, for large enough s, the equation

formula here

is solvable in positive integers x₁, …, x_s for sufficiently large integers n. The existence of such s for every f was established by Kamke [5] in 1921. Subsequent authors (Pillai, Hua [2–4], Vinogradov, Načaev [7], and others) have studied the problem of bounding G(f), the least s for which (1.1) is solvable for all large n.

Questions of local solubility of (1.1), that is, solubility of the congruence

formula here

play a more important and complicated role in this problem than in the classical Waring problem. Let Γ₀(f) denote the least number s so that (1.2) is solvable for every pair n, q. It is well known that Γ₀(x^k) [les ] 4k for every k, but Hua [4] found that for every k, the polynomial

formula here

satisfies Γ₀(f_k) [ges ] 2^k − 1 (take s = 2^k−2, q = 2^k and n = (−1)^k in (1.2)). Clearly G(f) [ges ] Γ₀(f), but one can say more by restricting the values of n under consideration, as has been done by several authors in the case f(x) = x⁴ (for example [1, 6]).

The singular series

formula here

where e(z) = e^2πiz, encapsulates the local solubility information. In particular, [Sfr ]_{s, f}(n) [ges ] 0 for every n and [Sfr ]_{s, f}(n) > 0 if and only if (1.2) is soluble for every q. Define G(f) to be the least number s so that for every δ >0 and every n>n₀(δ) with [Sfr ]_{s, f}(n) [ges ] δ, (1.1) is soluble. The reason for taking [Sfr ]_{s, f}(n) [ges ] δ instead of [Sfr ]_{s, f}(n) > 0 is that we wish to exclude from consideration certain n lying in sparse sequences for which (1.1) is insoluble but [Sfr ]_{s, f}(n) > 0. For example, taking f(x) = x⁴, s = 15 and n_j = 79·16^j (j = 0, 1, …), it can be shown that (1.1) is not soluble for n = n_j, that [Sfr ]_{s, f}(n_j) > 0 for all j, and that [Sfr ]_{s, f}(n_j) → 0 as j → ∞. It is known that G(x⁴) = 16 (see [1]) and that G(x⁴) [ges ] 11 almost holds (see [6]).