Quarterly Journal of Mathematics ( IF 0.704 ) Pub Date : 2020-02-03 , DOI: 10.1093/qmathj/haz052
Bo-Hae Im; Michael Larsen

Let |$f\in{\mathbb{Q}}(x)$| be a non-constant rational function. We consider ‘Waring’s problem for |$f(x)$|⁠,’ i.e., whether every element of |${\mathbb{Q}}$| can be written as a bounded sum of elements of |$\{f(a)\mid a\in{\mathbb{Q}}\}$|⁠. 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 |${\mathbb{Q}}$| can be represented as a bounded sum of elements of |$\{\pm f(a)\mid a\in{\mathbb{Q}}\}$|⁠.

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