An estimate of the number of terms in Waring's problem for polynomials of general form

A sharp upper bound is established for the smallest $s$ for which the equation $f(x_1)+\dots+f(x_s)=N$ is solvable in nonnegative integers $x_1,\dots,x_s$ for any fixed integer-valued polynomial $f(x)=a_n\binom xn+\dots+a_1\binom x1$ with $(a_n,\dots,a_1)=1$ and $a_n>0$ for all natural $N\geqslant N_0(f)$.
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