Signed sums of polynomial values

Abstract

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