WARING'S PROBLEM FOR RATIONAL FUNCTIONS IN ONE VARIABLE

Abstract

Let f is an element of Q(x) be a non-constant rational function. We consider 'Waring's problem for f(x),' i.e., whether every element of Q can be written as a bounded sum of elements of {f(a) vertical bar a is an element of Q}. For rational functions of degree 2, we give necessary and sufficient conditions. For higher degrees, we prove that every polynomial of odd degree and every odd Laurent polynomial satisfies Waring's problem. We also consider the 'easier Waring's problem': whether every element of Q can be represented as a bounded sum of elements of {+/- f(a) vertical bar a is an element of Q}.