Quartic polynomials in two variables do not represent all non-negative integers
Yao Xiao S, Yamagishi S. 2025. Quartic polynomials in two variables do not represent all non-negative integers. Journal of the European Mathematical Society.
Download (ext.)
Journal Article
| Epub ahead of print
| English
Author
Yao Xiao, Stanley;
Yamagishi, ShuntaroISTA
Corresponding author has ISTA affiliation
Department
Abstract
We prove that there does not exist F∈Q[x,y] of degree 4 such that F(Z^2 )=Z ≥0. In particular, this answers a question by John S. Lew and Bjorn Poonen for quartic polynomials.
Publishing Year
Date Published
2025-08-06
Journal Title
Journal of the European Mathematical Society
Publisher
EMS Press
Acknowledgement
The first author would like to thank Samir Siksek for introducing the problem
to him. The material in Section 6 is a result of discussing with many people, and the second author is very grateful to Tim Browning, Stephanie Chan, Jakob Glas, Jakub Löwit, Mirko Mauri, Marta Pieropan, Mike Roth, Matteo Verzobio and Victor Wang for taking the time to answer his questions and for their valuable suggestions. We thank Tim Browning, Yijie Diao, Ana Marija Vego and the anonymous referee for their helpful comments.
The first author was supported by NSERC Discovery Grant RGPIN-2024-06810. The
second author was supported by the NWO Veni Grant 016.Veni.192.047 during his time at
Utrecht University and by a FWF grant (DOI 10.55776/P32428) at the Institute of Science and
Technology Austria while working on this paper.
ISSN
eISSN
IST-REx-ID
Cite this
Yao Xiao S, Yamagishi S. Quartic polynomials in two variables do not represent all non-negative integers. Journal of the European Mathematical Society. 2025. doi:10.4171/jems/1697
Yao Xiao, S., & Yamagishi, S. (2025). Quartic polynomials in two variables do not represent all non-negative integers. Journal of the European Mathematical Society. EMS Press. https://doi.org/10.4171/jems/1697
Yao Xiao, Stanley, and Shuntaro Yamagishi. “Quartic Polynomials in Two Variables Do Not Represent All Non-Negative Integers.” Journal of the European Mathematical Society. EMS Press, 2025. https://doi.org/10.4171/jems/1697.
S. Yao Xiao and S. Yamagishi, “Quartic polynomials in two variables do not represent all non-negative integers,” Journal of the European Mathematical Society. EMS Press, 2025.
Yao Xiao S, Yamagishi S. 2025. Quartic polynomials in two variables do not represent all non-negative integers. Journal of the European Mathematical Society.
Yao Xiao, Stanley, and Shuntaro Yamagishi. “Quartic Polynomials in Two Variables Do Not Represent All Non-Negative Integers.” Journal of the European Mathematical Society, EMS Press, 2025, doi:10.4171/jems/1697.
All files available under the following license(s):
Copyright Statement:
This Item is protected by copyright and/or related rights. [...]
Link(s) to Main File(s)
Access Level
Open Access
Export
Marked PublicationsOpen Data ISTA Research Explorer
Sources
arXiv 2307.05712
Google Scholar