Signed sums of polynomial values

Abstract:

We give a generalization of Bleicher’s result on signed sums of $k$th powers. Let $f(x)$ be an integral-valued polynomial of degree $k$ satisfying the necessary condition that there exists no integer $d>1$ dividing the values $f(x)$ for all integers $x$. Then, for every positive integer $n$ and every integer $l$, there are infinitely many integers $m\ge l$ and choices of $\varepsilon _{i}=\pm 1$ such that \[ n=\sum _{i=l}^{m}\varepsilon _{i}f(i).\]