Waring's problem for polynomials of small degree (Q1080464)

Let \(n\geq 2\), n an integer, and \(f(x)=a_ nx^ n+...+a_ 1x\) a polynomial with integer coefficients \((a_ n,a_{n-1},...,a_ 1)=1\), \(a_ n>0\). Let \(d=\gcd f(x)\) denote the greatest common divisor of the values of the polynomial f(x) for x integer. Let G(f) be the least integer r with the following properties: there exists a constant c with the condition that for \(r=G(f)\) every integer \(N\geq c\) can be represented in the form \[ (1)\quad N=(f(x_ 1)+f(x_ 2)+...+f(x_ r))/d \] with integer nonnegative \(x_ i\), but there is no \(c'>c\) such that every integer \(N\geq c'\) can be represented in the form (1) for \(r<G(f).\) For polynomials of small degree of a certain class \textit{V. I. Nechaev} [Tr. Mat. Inst. Steklova 38, 190-243 (1951; Zbl 0053.359)] has obtained estimates for G(f). In the present paper an improvement of these estimates is obtained.