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

Stanley Yao Xiao; Shuntaro Yamagishi

doi:10.4171/jems/1697

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

Stanley Yao Xiao
University of Northern British Columbia, Prince George, British Columbia, Canada
ORCID
Shuntaro Yamagishi
IST Austria, Klosterneuburg, Austria
ORCID

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Abstract

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

Cite this article

Stanley Yao Xiao, Shuntaro Yamagishi, Quartic polynomials in two variables do not represent all non-negative integers. J. Eur. Math. Soc. (2025), published online first

DOI 10.4171/JEMS/1697