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, …, xs
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(xk) [les ] 4k
for every k, but Hua [4] found that for every k, the polynomial
formula here
satisfies Γ0(fk) [ges ] 2k − 1 (take
s = 2k−2, q = 2k 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) = e2πiz, 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 nj = 79·16j
(j = 0, 1, …), it can be shown that (1.1) is not soluble for
n = nj, that [Sfr ]s, f(nj) > 0
for all j, and that
[Sfr ]s, f(nj) → 0 as j → ∞.
It is known that G(x4) = 16 (see [1]) and that
G(x4) [ges ] 11 almost holds (see [6]).
