On Waring's Problem with Cubic Polynomial Summands

Let a cubic integral-valued polynomial be represented by

P(x) = 1/6a(x³-x)+1/6b(x²-x)+cx+d,

where a, b, c and d are integers with (a, b, c) = 1 and a > 0. The object of this paper is to prove that the Diophantine equation

P(x₁) + ...+P(x<sub8)=N x_n>0

is soluble for all sufficiently large integers N. This result is better than my previous one, where we require nine values of P(x), x ≥ 0.