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Waring's problem for rational functions in one variable

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Bo-Hae Im, Michael Larsen

Number Theory · Submitted 21 Jan 2018

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim For higher degrees, we prove that every polynomial of odd degree and every odd Laurent polynomial satisfies Waring's Problem.

how it is stated · asserted 1.00 read the record →

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arXiv ID1801.06770 (version 1)
Pith IDWZM3NOU6, Pith's permanent record of this paper and its review
CitationBo-Hae Im, Michael Larsen. "Waring's problem for rational functions in one variable." arXiv:1801.06770 (2018). https://pith.science/paper/WZM3NOU6
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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,285 input tokens.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.45 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.15 · compares with prior results 0.08 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

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

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 Let $f\in \mathbb{Q}(x)$ be a non-constant rational function. background 0.99 0.08 0.01 asserted 0.64
s1 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 $\{f(a)\mid a\in \mathbb{Q}\}$. background 0.76 0.23 0.05 unclear 0.78
s2 For rational functions of degree $2$, we give necessary and sufficient conditions. result 0.82 0.94 0.17 asserted 1.00
s3 For higher degrees, we prove that every polynomial of odd degree and every odd Laurent polynomial satisfies Waring's Problem. result 0.99 0.95 0.35 asserted 1.00
s4 We also consider the "Easier Waring's Problem": whether every element of $\mathbb{Q}$ can be represented as a bounded sum of elements of $\{\pm f(a)\mid a\in \mathbb{Q}\}$. contribution 0.92 0.81 0.04 unclear 0.74

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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  • T body sentences
  • C citations
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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original 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 $\{\pm f(a)\mid a\in \mathbb{Q}\}$.

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What this paper does with the arXiv papers it cites

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

What it cites

Cites as background 1 paper
  • This paper was written simultaneously with the paper [ref] of the second named author and Dong Quan Ngoc Nguyen on Waring's Problem for unipotent algebraic groups over number fields.

    Waring's problem for unipotent algebraic groups · Introduction · 0.92

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