Quartic polynomials in two variables do not represent all non-negative integers

Stanley Yao Xiao; Shuntaro Yamagishi

Quartic polynomials in two variables do not represent all non-negative integers

Number Theory 2023-07-14 v2

Authors: Stanley Yao Xiao , Shuntaro Yamagishi

Abstract

In this paper, we prove that there does not exist $F \in \mathbb{Q}[x,y]$ of degree $4$ such that $F(\mathbb{Z}^2) = \mathbb{Z}_{\geq 0}$ . In particular, this answers a question by John S. Lew and Bjorn Poonen for quartic polynomials.

Cite

@article{arxiv.2307.05712,
  title  = {Quartic polynomials in two variables do not represent all non-negative integers},
  author = {Stanley Yao Xiao and Shuntaro Yamagishi},
  journal= {arXiv preprint arXiv:2307.05712},
  year   = {2023}
}

Comments

We made a few changes in the wording and fixed typos. 25 pages. See link for Poonen's question on MathOverflow: https://mathoverflow.net/questions/9731/polynomial-representing-all-nonnegative-integers

Quartic polynomials in two variables do not represent all non-negative integers

Abstract

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