Waring's problem for rational functions in one variable

TL;DR

This paper investigates Waring's problem for rational functions over the rationals, establishing conditions for degree 2 and proving results for higher degrees, including odd degree polynomials.

Contribution

It provides necessary and sufficient conditions for degree 2 rational functions and proves that all odd degree polynomials satisfy Waring's problem over rationals.

Findings

Degree 2 case characterized by specific conditions

All odd degree polynomials satisfy Waring's problem

Analysis of the 'Easier Waring's Problem' variant

Abstract

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 $\bbq$ 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 $\{\pm f(a)\mid a\in \mathbb{Q}\}$.